Research ArticleAveraging Principles for Nonautonomous Two-Time-ScaleStochastic Reaction-Diffusion Equations with Jump

Yong Xu 12 and Ruifang Wang1

1Department of Applied Mathematics Northwestern Polytechnical University Xirsquoan 710072 China2MIIT Key Laboratory of Dynamics and Control of Complex Systems Northwestern Polytechnical UniversityXirsquoan 710072 China

Correspondence should be addressed to Yong Xu hsux3nwpueducn

Received 9 January 2020 Revised 7 July 2020 Accepted 14 July 2020 Published 19 September 2020

Academic Editor Eulalia Martınez

Copyright copy 2020 Yong Xu and Ruifang Wang is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we aim to develop the averaging principle for a slow-fast system of stochastic reaction-diffusion equations driven byPoisson randommeasures e coefficients of the equation are assumed to be functions of time and some of them are periodic oralmost periodic erefore the Poisson term needs to be processed and a new averaged equation needs to be given For thisreason the existence of time-dependent evolution family of measures associated with the fast equation is studied and proved thatit is almost periodic Next according to the characteristics of almost periodic functions the averaged coefficient is defined by theevolution family of measures and the averaged equation is given Finally the validity of the averaging principle is verified by usingthe Khasminskii method

1 Introduction

e slow-fast systems are widely encountered in biologyecology and other application areas In this paper we are

concerned with the following nonautonomous slow-fastsystems of stochastic partial differential equations (SPDEs)on a bounded domain O of Rd(dge 1)

zuϵzt

(t ξ) A1(t)uϵ(t ξ) + b1 t ξ uϵ(t ξ) vϵ(t ξ)( 1113857 + f1 t ξ uϵ(t ξ)( 1113857zωQ1

zt(t ξ) + 1113946

Zg1 t ξ uϵ(t ξ) z( 1113857

z 1113957N1

zt(t ξ dz)

zvϵzt

(t ξ) 1ϵ

A2(t) minus α( 1113857vϵ(t ξ) + b2 t ξ uϵ(t ξ) vϵ(t ξ)( 11138571113858 1113859 +1ϵ

radic f2 t ξ uϵ(t ξ) vϵ(t ξ)( 1113857zωQ2

zt(t ξ) + 1113946

Zg2 t ξ uϵ(t ξ) vϵ(t ξ) z( 1113857

z 1113957Nϵ2

zt(t ξ dz)

uϵ(0 ξ) x(ξ)

vϵ(0 ξ) y(ξ)

ξ isin O

N1uϵ(t ξ) N2vϵ(t ξ) 0 tge 0 ξ isin zO

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

HindawiComplexityVolume 2020 Article ID 9864352 22 pageshttpsdoiorg10115520209864352

where ϵ≪ 1 is a positive parameter and α is a sufficientlylarge fixed constante operatorsN1 andN2 are boundaryoperatorse stochastic perturbations ωQ1 ωQ2 and 1113957N1

1113957Nϵ2

are mutually independent Wiener processes and Poissonrandom measures on the same complete stochastic basis(ΩF Ft1113864 1113865tge0P) which will be described in Section 2 Fori 1 2 the operator Ai(t) and the functions bi fi andgi

depend on time and we assume that the operator A2(t) isperiodic and the functions b1 b2 f2 andg2 are almostperiodic

e goal of this paper is to establish an effective ap-proximation for the slow equation of original system (1) byusing the averaging principle e averaged equation isobtained as follows

zu

zt(t ξ) A1(t)u(t ξ) + B1(u(t))(ξ) + f1(t ξ u(t ξ))

zωQ1

zt(t ξ) + 1113946

Zg1(t ξ u(t ξ) z)

z 1113957N1

zt(t ξ dz)

u(0 ξ) x(ξ) ξ isin O

N1u(t ξ) 0 tge 0 ξ isin zO

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(2)

where B1 is the averaged coefficient which will be given inequation (6) To demonstrate the validity of the averagingprinciple we prove that for any Tgt 0 and ηgt 0 it yields

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

L2(O)gt η1113888 1113889 0 (3)

where u is the solution of the averaged equation (2)e theory of the averaging principle originated by Laplace

and Lagrange has been applied in celestial mechanics oscil-lation theory radiophysics and other fieldse firstly rigorousresults for the deterministic case were given by Bogolyubov andMitropolskii [1] Moreover Volosov [2] and Besjes [3] alsopromoted the development of the averaging principle engreat interest has appeared in its application to dynamicalsystems under random perturbations An important contri-bution was that in 1968 Khasminskii [4] originally proposedthe averaging principle for stochastic differential equations(SDEs) driven by Brownian motion Since then the averagingprinciple has been an active area of research Many studies onthe averaging principle of SDEs have been presented eg Givon[5] Freidlin andWentzell [6] Duan [7]ompson [8] andXuand his coworkers [9ndash11] Recently effective approximationsfor slow-fast SPDEs have been received extensive attentionCerrai [12 13] investigated the validity of the averagingprinciple for a class of stochastic reaction-diffusion equationswith multiplicative noise In addition Wang and Roberts [14]Pei and Xu [15ndash17] and Li [18] and Gao [19ndash21] also studiedthe averaging principles for the slow-fast SPDEs

e abovementioned works mainly considered autono-mous systems For the autonomous systems as long as theinitial value is given the solution of which only depends on the

duration of time not on the selection of the initial timeHowever if the initial time is different the solution of non-autonomous equations with the same initial data will also bedifferent erefore compared with the autonomous systemsthe dynamic behaviors of the nonautonomous systems aremore complex which can portray more actual models Che-pyzhov and Vishik [22] studied long-time dynamic behaviorsof nonautonomous dissipative system Carvalho [23] dealt withthe theory of attractors for the nonautonomous dynamicalsystems Bunder and Roberts [24] considered the discretemodelling of nonautonomous PDEs

In 2017 the averaging principle has been presented fornonautonomous slow-fast system of stochastic reaction-diffusion equations by Cerrai [25] But the system of thispaper was driven by Gaussian noises which is considered asan ideal noise source and only can simulate fluctuations nearthe mean value Actually due to the complexity of the ex-ternal environment random noise sources encountered inpractical fields usually exhibit non-Gaussian propertieswhich may cause sharp fluctuations It should be pointed outthat Poisson noise one of the most ubiquitous noise sourcesin many fields [26ndash28] can provide an accurate mathe-matical model to describe discontinuous random processessome large moves and unpredictable events [29ndash31] So inthis paper we are devoted to developing the averagingprinciple for nonautonomous systems of reaction-diffusionequations driven by Wiener processes and Poisson randommeasures

e key to using the averaging principle to analyzesystem (1) is the fast equation with a frozen slow componentx isin L2(O)

zvxy

zt(t ξ) A2(t) minus α( 1113857v

xy(t ξ) + b2 t ξ x(ξ) v

xy(t ξ)( 11138571113858 1113859 + f2 t ξ x(ξ) v

xy(t ξ)( 1113857

zωQ2

zt(t ξ) + 1113946

Zg2 t ξ x(ξ) v

xy(t ξ) z( 1113857

z 1113957Nϵ2

zt(t ξ dz)

vxy(s ξ) y(ξ) ξ isin O

N2vxy(t ξ) 0 tge s ξ isin zO

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(4)

2 Complexity

By dealing with the Poisson terms we prove that anevolution family of measures (μx

t t isin R) on L2(O) for fastequation (4) exists en assuming thatA2(t) is periodic andb2 f2 andg2 are almost periodic we prove that the evolutionfamily ofmeasures is almost periodicMoreover with the aid ofeorem 210 in [32] we prove that the family of functions

t isin R⟼1113946L2(O)

B1(t x y)μxt (dy)1113896 1113897 (5)

is uniformly almost periodic for any x in a compact setK sub L2(O) where B1(t x y)(ξ) b1(t ξ x(ξ) y(ξ)) forany x y isin L2(O) and ξ isin O

According to the characteristics of almost periodicfunction ([25] eorem 34) we define the averaged co-efficient B1 as follows

B1(x) ≔ limT⟶infin

1T

1113946T

01113946

L2(O)B1(t x y)μx

t (dy)dt x isin L2(O)

(6)

Finally the averaged equation is obtained through theaveraged coefficient B1 Using the classical Khasminskiimethod to the present situation the averaging principle iseffective

e abovementioned notations will be given in Section 2and in this paper cgt 0 below with or without subscripts willrepresent a universal constant whose value may vary indifferent occasions

2 Notations Assumptions and Preliminaries

Let O be a bounded domain of Rd(dge 1) having a smoothboundary In this paper we denote H the separable Hilbertspace L2(O) endowed with the usual scalar product

langx yrangH 1113946O

x(ξ)y(ξ)dξ (7)

and with the corresponding norm middotH e norm in Linfin(O)

will be denoted by middotinfinFurthermore the subspace D((minus A)θ) [33ndash35] of the

generator A is dense in H and endowed with the norm

Λθ (minus A)θΛ

H Λ isin D (minus A)

θ1113872 1113873 (8)

for 0le θlt 1 0lt tleT According to eorem 613 in [36]there exists cθ gt 0 such that

(minus A)θe

At

Hle cθt

minus θ (9)

We denote by Bb(H) the Banach space of the boundedBorel functions φ H⟶ R endowed with the sup-norm

φ0 ≔ supxisinH

|φ(x)| (10)

and Cb(H) is the subspace of the uniformly continuousmappings Moreover D([s T]H) denotes the space of allcadlag path from [s T] into H

We shall denote that L(H) is the space of the boundedlinear operators in H and denote L2(H) the subspace ofHilbertndashSchmidt operators endowed with the norm

Q2 Tr QlowastQ[ ]

1113968 (11)

In slow-fast system (1) the Gaussian noises zωQ1 zt(t ξ)

and zωQ2 zt(t ξ) are assumed to be white in time andcolored in space in the case of space dimension dgt 1 fortge 0 and ξ isin O And ωQi (t ξ)(i 1 2) is the cylindricalWiener processes defined as

ωQi (t ξ) 1113944infin

k1Qiek(ξ)βk(t) i 1 2 (12)

where ek1113864 1113865kisinN is a complete orthonormal basis in Hβk(t)1113864 1113865kisinN is a sequence of mutually independent standardBrownian motion defined on the same complete stochasticbasis (ΩF Ft1113864 1113865tge0P) and Qi is a bounded linear operatoron H

Next we give the definitions of Poisson random mea-sures 1113957N1(dt dz) and 1113957N

ϵ2(dt dz) Let (ZB(Z)) be a given

measurable space and v(dz) be a σ-finite measure on itDpi

t i 1 2 represents two countable subsets of R+

Moreover let p1t t isin Dp1

tbe a stationaryFt-adapted Poisson

point process on Z with the characteristic v and p2t t isin Dp2

t

be the other stationaryFt-adapted Poisson point process onZ with the characteristic vϵ Denote by Ni(dt dz) i 1 2the Poisson counting measure associated with pi

t ie

Ni(tΛ) ≔ 1113944sisinD

pitslet

IΛ pit1113872 1113873 i 1 2

(13)

Let us denote the two independent compensated Poissonmeasures

1113957N1(dt dz) ≔ N1(dt dz) minus v1(dz)dt

1113957Nϵ2(dt dz) ≔ N2(dt dz) minus

1ϵv2(dz)dt

(14)

where v1(dz)dt and 1ϵv2(dz)dt are the compensatorsRefer to [28 37] for a more detailed description of the

stochastic integral with respect to a cylindrical Wienerprocess and Poisson random measure

For any t isin R the operators A1(t) and A2(t) are sec-ond-order uniformly elliptic operators having continuouscoefficients on O e operators N1 and N2 are theboundary operators which can be either the identity op-erator (Dirichlet boundary condition) or a first-order op-erator (coefficients satisfying a uniform nontangentialitycondition) We shall assume that the operatorAi(t) has thefollowing form

Ai(t) ci(t)Ai + Li(t) t isin R i 1 2 (15)

where Ai is a second order uniformly elliptic operator[38 39] with continuous coefficients on O which is inde-pendent of t In addition Li(t) is a first-order differentialoperator which has the form

Li(t ξ)u(ξ) langli(t ξ)nablau(ξ)rangRd t isin R ξ isin O (16)

e realizations of the differential operators Ai and Li

in H are Ai and Li Moreover A1 and A2 generate twoanalytic semigroups etA1 and etA2 respectively

Complexity 3

Now we give the following assumptions

(A1)

(a) For i 1 2 the function ci R⟶ R is con-tinuous and there exist c0 cgt 0 such that

c0 le ci(t)le c t isin R (17)

(b) For i 1 2 the function li R times O⟶ Rd iscontinuous and bounded

(A2) For i 1 2 there exists a complete orthonormalsystem eik1113966 1113967

kisinN in H and two sequences of nonneg-ative real numbers αik1113966 1113967

kisinN and λik1113966 1113967kisinN such that

Aieik minus αikeik

Qieik λikeik kge 1

κi ≔ 1113944infin

k1λρi

ik eik

2infin ltinfin

ζ i ≔ 1113944infin

k1αminus βi

ik eik

2infin ltinfin

(18)

for some constants ρi isin (2 +infin] and βi isin (0 +infin)

such that

βi ρi minus 2( 11138571113858 1113859

ρi

lt 1 (19)

Remark 1 For more comments and examples about theassumption (A2) of the operators Ai and Qi the reader canread [12]

(A3) e mappings b1 R times O times R2⟶ R f1 R times O times

R⟶ R g1 R times O times R times Z⟶ R are measur-able and the mappings b1(t ξ middot) R2⟶ R

f1(t ξ middot) R⟶ R g1(t ξ middot z) R⟶ R areLipschitz continuous and linearly growing uni-formly with respect to (t ξ z) isin R times O times Z More-over for all pge 1 there exist positive constantsc1 and c2 such that for all x1 x2 isin R we have

sup(tξ)isinRtimesO

1113946Z

g1 t ξ x1 z( 11138571113868111386811138681113868

1113868111386811138681113868pυ1(dz)

le c1 1 + x11113868111386811138681113868

1113868111386811138681113868p

1113872 1113873

sup(tξ)isinRtimesO

1113946Z

g1 t ξ x1 z( 1113857 minus g1 t ξ x2 z( 11138571113868111386811138681113868

1113868111386811138681113868pυ1(dz)

le c2 x1 minus x21113868111386811138681113868

1113868111386811138681113868p

(20)

(A4) e mappings b2 R times O times R2⟶ R f2 R times O times

R2⟶ R g2 R times O times R2 times Z⟶ R are mea-surable and the mappings b2(t ξ middot) R2⟶ R

f2(t ξ middot) R2⟶ R g2(t ξ middot z) R2⟶ R areLipschitz continuous and linearly growing uni-formly with respect to (t ξ z) isin R times O times Z

Moreover for all qge 1 there exist positive con-stants c3 and c4 such that for all(xi yi) isin R2 i 1 2 we have

sup(tξ)isinRtimesO

1113946Z

g2 t ξ x1 y1 z( 11138571113868111386811138681113868

1113868111386811138681113868qυ2(dz)

le c3 1 + x11113868111386811138681113868

1113868111386811138681113868q

+ y11113868111386811138681113868

1113868111386811138681113868q

1113872 1113873

sup(tξ)isinRtimesO

1113946Z

g2 t ξ x1 y1 z( 11138571113868111386811138681113868

minus g2 t ξ x2 y2 z( 1113857|qυ2(dz)

le c4 x1 minus x21113868111386811138681113868

1113868111386811138681113868q

+ y1 minus y21113868111386811138681113868

1113868111386811138681113868q11138731113872 (21)

Remark 2 For any (t ξ) isin R times O and x y h isin H z isin Z weshall set

B1(t x y)(ξ) ≔ b1(t ξ x(ξ) y(ξ))

B2(t x y)(ξ) ≔ b2(t ξ x(ξ) y(ξ))

F1(t x)h1113858 1113859(ξ) ≔ f1(t ξ x(ξ))h(ξ)

F2(t x y)h1113858 1113859(ξ) ≔ f2(t ξ x(ξ) y(ξ))h(ξ)

G1(t x z)h1113858 1113859(ξ) ≔ g1(t ξ x(ξ) z)h(ξ)

G2(t x y z)h1113858 1113859(ξ) ≔ g2(t ξ x(ξ) y(ξ) z)h(ξ)

(22)

Due to (A3) and (A4) for any fixed (t z) isin (RZ) themappings

B1(t middot) H times H⟶ H B2(t middot) H times H⟶ H

F1(t middot) H⟶L(H) F2(t middot) H times H⟶L(H)

G1(t middot z) H⟶L(H) G2(t middot z) H times H⟶L(H)

(23)

are Lipschitz continuous and have linear growth conditionsNow for i 1 2 we define

ci(t s) ≔ 1113946t

sci(r)dr slt t (24)

and for any ϵgt 0 and βge 0 set

Uβϵi(t s) e1ϵci(ts)Ai minus βϵ(tminus s)

slt t (25)

For ϵ 1 we write Uβi(t s) and for ϵ 1 and β 0 wewrite Ui(t s)

Next for any ϵgt 0 βge 0 and for anyu isin C([s t] W

1p0 (O)) we define

ψβϵi(u s)(r) 1ϵ

1113946r

sUβϵi(r ρ)Li(ρ)u(ρ)dρ slt rlt t

(26)

In the case ϵ 1 we write ψβi(u s)(r) and in the caseϵ 1 and β 0 we write ψi(u s)(r)

Remark 3 By using the same argument as Chapter 5 of thebook [40] we can get that there exists a unique evolution

system U(t s) eA 1113938

t

sc(r)dr

(0le sle tleT) for c(t)A More-over we also can get that ψβϵi(u s)(t) is the solution of

4 Complexity

du(t) 1ϵ

Ai(t) minus β( 1113857u(t)dt tgt s u(s) 0 (27)

3 A Priori Bounds for the Solution

With all notations introduced above system (1) can berewritten in the following abstract form

duϵ(t) A1(t)uϵ(t) + B1 t uϵ(t) vϵ(t)( 11138571113858 1113859dt

+ F1 t uϵ(t)( 1113857dωQ1(t)

+ 1113946Z

G1 t uϵ(t) z( 1113857 1113957N1(dt dz)

dvϵ(t) 1ϵ

A2(t) minus α( 1113857vϵ(t) + B2 t uϵ(t) vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ(t) vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ(t) vϵ(t) z( 1113857 1113957Nϵ2(dt dz)

uϵ(0) x vϵ(0) y

(28)

According to Remark 2 we know that the coefficients ofsystem (28) satisfy global Lipschitz and linear growthconditions and the assumptions (A1)ndash(A4) are uniformwith respect to t isin R So using the same argument as[28 41 42] it is easy to prove that for any ϵgt 0 Tgt 0 andx y isin H there exists a unique adapted mild solution (uϵ vϵ)

to system (28) in Lp(Ω D([0 T]H times H)) is means thatthere exist two unique adapted processes uϵ and vϵ inLp(Ω D([0 T]H)) such that

uϵ(t) U1(t 0)x + ψ1 uϵ 0( 1113857(t)

+ 1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

vϵ(t) Uαϵ2(t 0)y + ψαϵ2 vϵ 0( 1113857(t)

+1ϵ

1113946t

0Uαϵ2(t r)B2 r uϵ(r) vϵ(r)( 1113857dr

+1ϵ

radic 1113946t

0Uαϵ2(t r)F2 r uϵ(r) vϵ(r)( 1113857dw

Q2(r)

+ 1113946t

01113946Z

Uαϵ2(t r)G2 r uϵ(r) vϵ(r) z( 1113857 1113957Nϵ2(dr dz)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

Lemma 1 Under (A1)ndash(A4) for any pge 1 and Tgt 0 thereexists a positive constant cpT such that for any x y isin H andϵ isin (0 1] we have

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873 (30)

1113946T

0E vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (31)

Proof For fixed ϵ isin (0 1] and x y isin H for any t isin [0 T]we denote

Γ1ϵ(t) ≔ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

Ψ1ϵ(t) ≔ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

(32)

Setting Λ1ϵ(t) ≔ uϵ(t) minus Γ1ϵ(t) minus Ψ1ϵ(t) we have

ddtΛ1ϵ(t) c1(t)A1Λ1ϵ(t) + L1(t) Λ1ϵ(t) + Γ1ϵ(t)1113872

+ Ψ1ϵ(t)1113873B1 tΛ1ϵ(t) + Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873

Λ1ϵ(0) x

(33)

For any pge 2 because B1(middot) is Lipschitz continuoususing Youngrsquos inequality we have1p

ddtΛ1ϵ(t)

p

Hlangc1(t)A1Λ1ϵ(t)Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langL1(t) Λ1ϵ(t) + Γ1ϵ(t) + Ψ1ϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langB1 tΛ1ϵ(t) + Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873

minus B1 t Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langB1 t Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873

minus B1 t Ψ1ϵ(t) vϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langB1 t Ψ1ϵ(t) vϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

le c Λ1ϵ(t)

p

H+ c Γ1ϵ(t)

H Λ1ϵ(t)

pminus 1H

+ c Ψ1ϵ(t)

H Λ1ϵ(t)

pminus 1H

+ c B1 t Ψ1ϵ(t) vϵ(t)1113872 1113873

HΛ1ϵ(t)

pminus 1H

le cp Λ1ϵ(t)

p

H+ cp 1 + Γ1ϵ(t)

p

H+ Ψ1ϵ(t)

p

H+ vϵ(t)

p

H1113872 1113873

(34)

is implies that

Λ1ϵ(t)

p

Hle e

cptx

p

H + cp 1113946t

0e

cp(tminus r) 1 + Γ1ϵ(r)

p

H1113872

+ Ψ1ϵ(r)

p

H+ vϵ(r)

p

H1113873dr

(35)

Complexity 5

According to the definition of Λ1ϵ(t) for any t isin [0 T]we have

uϵ(t)

p

Hle cpT 1 +x

p

H + suprisin[0T]

Γ1ϵ(r)

p

H+ sup

risin[0T]

Ψ1ϵ(r)

p

H1113888 1113889

+ cpT 1113946T

0vϵ(r)

p

Hdr

(36)

So

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H + E suptisin[0T]

Γ1ϵ(t)

p

H1113888

+ E suptisin[0T]

Ψ1ϵ(t)

p

H1113889 + cpT 1113946

T

0E vϵ(r)

p

Hdr

(37)

According to Lemma 41 in [12] with θ 0 it is easy toprove that

E suptisin[0T]

Γ1ϵ(t)p

H le cpT 1113946T

01 + Euϵ(r)

p

H1113872 1113873dr (38)

Due to (A3) using Kunitarsquos first inequality we get

E suptisin[0T]

Ψ1ϵ(t)

p

Hle cpE 1113946

T

01113946Z

ec1(tr)A1G1 r uϵ(r) z( 1113857

2

Hv1(dz)dr1113888 1113889

p2

+ cpE1113946T

01113946Z

ec1(tr)A1G1 r uϵ(r) z( 1113857

p

Hv1(dz)dr

le cpE 1113946T

01 + uϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+ cpE1113946T

01 + uϵ(r)

p

H1113872 1113873drle cpT 1113946

T

01 + E uϵ(r)

p

H1113872 1113873dr

(39)

E suptisin[0T]

Ψ1ϵ(t)

p

Hle cpT 1113946

T

01 + E uϵ(r)

p

H1113872 1113873dr (40)

Substituting (38) and (39) into (37) we obtain

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H + 1113946T

0E vϵ(r)

p

Hdr1113888 1113889

+ cpT 1113946T

0E sup

σisin[0r]

uϵ(σ)

p

Hdr

(41)

Now we have to estimate

1113946T

0E vϵ(r)

p

Hdr (42)

For any t isin [0 T] we set

Γ2ϵ(t) ≔1ϵ

radic 1113946t

0Uαϵ2(t r)F2 r uϵ(r) vϵ(r)( 1113857dw

Q2(r)

Ψ2ϵ(t) ≔ 1113946t

01113946Z

Uαϵ2(t r)G2 r uϵ(r) vϵ(r) z( 1113857 1113957Nϵ2(dr dz)

(43)

Let Λ2ϵ(t) ≔ vϵ(t) minus Γ2ϵ(t) minus Ψ2ϵ(t) we have

ddtΛ2ϵ(t)

1ϵ

c2(t)A2 minus α( 1113857Λ2ϵ(t) +1ϵL2(t)

middot Λ2ϵ(t) + Γ2ϵ(t) + Ψ2ϵ(t)1113872 1113873

+1ϵB2 t uϵ(t)Λ2ϵ(t) + Γ2ϵ(t)1113872

+ Ψ2ϵ(t)1113873 Λ2ϵ(0) y

(44)

For any pge 2 because αgt 0 is large enough by pro-ceeding as in (34) we can get

1p

ddtΛ2ϵ(t)

p

Hle minus

α2ϵΛ2ϵ(t)

p

H

+cp

ϵ1 + uϵ(t)

p

H+ Γ2ϵ(t)

p

H+ Ψ2ϵ(t)

p

H1113872 1113873

(45)

According to the Gronwall inequality we have

Λ2ϵ(t)

p

Hle

cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r)

middot 1 + uϵ(r)

p

H+ Γ2ϵ(r)

p

H+ Ψ2ϵ(r)

p

H1113872 1113873dr + e

minus (αp2ϵ)ty

p

H

(46)

6 Complexity

According to the definition of Λ2ϵ(t) for any t isin [0 T]we have

E vϵ(t)

p

Hle cpE Γ2ϵ(t)

p

H+ cpE Ψ2ϵ(t)

p

H+ cpe

minus (αp2ϵ)ty

p

H

+cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r) 1 + E uϵ(r)

p

H+ E Γ2ϵ(r)

p

H1113872

+ E Ψ2ϵ(r)

p

H1113873dr

(47)

erefore by integrating with respect to t using Youngrsquosinequality we obtain

1113946t

0E vϵ(r)

p

Hdrle cp(t)y

p

H + cp 1113946t

0E Γ2ϵ(r)

p

Hdr + 1113946

t

0E Ψ2ϵ(r)

p

Hdr1113888 1113889 +

cp

ϵ1113946

t

01 + E uϵ(r)

p

H+ E Γ2ϵ(r)

p

H+ E Ψ2ϵ(r)

p

H1113872 1113873dr

times 1113946t

0e

minus (αp2ϵ)rdrle cp(t) 1 +yp

H1113872 1113873 + cp 1113946t

0E uϵ(r)

p

Hdr + 1113946

t

0E Γ2ϵ(r)

p

Hdr + 1113946

t

0E Ψ2ϵ(r)

p

Hdr1113888 1113889

(48)

According to the BurkholderndashDavisndashGundy inequalityby proceeding as Proposition 42 in [12] we can easily get

1113946t

0E Γ2ϵ(r)

p

Hdrle cp(t) 1113946

t

01 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(49)

Concerning the stochastic term Ψ2ϵ(t) using Kunitarsquosfirst inequality we have

E Ψ2ϵ(t)

p

Hle cpE

1ϵ

1113946t

01113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2G2 r uϵ(r) vϵ(r) z( 1113857

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

01113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2G2 r uϵ(r) vϵ(r) z( 1113857

p

Hv2(dz)dr

lecp

ϵp2E1113946

t

0e

minus (α(pminus 2)ϵ)(tminus r) 1 + uϵ(r)

p

H+ vϵ(r)

p

H1113872 1113873dr 1113946

t

0e

minus (αpϵ)(pminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵE1113946

t

0e

minus (αpϵ)(tminus r) 1 + uϵ(r)

p

H+ vϵ(r)

p

H1113872 1113873drle

cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r) 1 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(50)

By integrating with respect to t both sides and usingYoungrsquos inequality we have

1113946t

0E Ψ2ϵ(r)

p

Hdrle

cp

ϵ1113946

t

0e

minus (αp2ϵ)rdr middot 1113946t

01 + E uϵ(r)

p

H1113872

+ E vϵ(r)

p

H1113873drle cp(t) 1113946

t

01 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(51)

Substituting (49) and (51) into (48) we get

1113946t

0E vϵ(r)

p

Hdrle cp(t) 1 +y

p

H + 1113946t

0E uϵ(r)

p

Hdr1113888 1113889

+ cpT(t) 1113946t

0E vϵ(r)

p

Hdr

(52)

As cp(0) 0 and cp(t) is a continuous increasingfunction we can fix t0 gt 0 such that for any tle t0 we havecp(t)le 12 so

Complexity 7

1113946t

0E vϵ(r)

p

Hdrle cp(t) 1 +y

p

H + E suprisin[0t]

uϵ(r)

p

H1113888 1113889 t isin 0 t01113858 1113859

(53)

Due to (41) and (53) for any t isin [0 t0] we can get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT(t)E suprisin[0t]

uϵ(r)

p

H+ cpT 1113946

t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr

(54)

Similarly we also can fix 0lt t1 le t0 such that for anytle t1 we have cpT(t)le 12 so

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT 1113946t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr t isin 0 t11113858 1113859

(55)

According to the Gronwall inequality we get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(56)

Substituting (56) into (53) it yields

1113946t

0E vϵ(r)

p

Hdrle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(57)

For any pge 2 by repeating this in the intervals[t1 2t1] [2t1 3t1] etc we can easily get (31) Substituting(31) into (41) and using the Gronwall inequality again we get(30) Using the Holder inequality we can estimate (30) and(31) for p 1

Lemma 2 Under (A1)ndash(A4) there exists θgt 0 such that forany Tgt 0 pge 1 x isin D((minus A1)

θ) with θ isin [0 θ) and y isin Hthere exist a positive constant cpθT gt 0 such that

supϵisin(01]

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (58)

Proof Assuming that x isin D((minus A1)θ)(θ ge 0) for any

t isin [0 T] we have

uϵ(t) U1(t 0)x + ψ1 uϵ 0( 1113857(t)

+ 1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

(59)

Concerning the second term ψ1(uϵ 0)(t) we get

ψ1 uϵ 0( 1113857(t)

p

θ le cp 1113946t

0minus A1( 1113857

θe

c1(tr)A1L1(r)uϵ(r)dr

p

H

le cpθ 1113946t

0(t minus r)

minus θL1(r)uϵ(r)

Hdr1113888 1113889

p

le cpθ suprisin[0T]

uϵ(r)

p

H1113946

t

0(t minus r)

minus θdr1113888 1113889

p

le cpθT 1 +xp

H +yp

H1113872 1113873

(60)

For any pge 2 according to the proof of Proposition 43in [12] and thanks to (30) and (31) it is possible to show thatthere exists a 1113957θ ge 0 such that for any θle 1113957θand12 we have

E suptisin[0T]

1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

p

θle cpθT 1 +x

p

H +yp

H1113872 1113873

(61)

E suptisin[0T]

Γ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (62)

Concerning the stochastic term Ψ1ϵ(t) using the fac-torization argument we have

Ψ1ϵ(t) cθ 1113946t

0(t minus r)

θminus 1e

c1(tr)A1ϕϵθ(r)dr (63)

where

ϕϵθ(r) 1113946r

01113946Z

(r minus σ)minus θ

ec1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

(64)

Next for any pge 2 let 1113954θ ((14) minus (12p))and12p forany θle 1113954θ according to (A3) and Lemma 1 using Kunitarsquosfirst inequality and the Holder inequality we get

8 Complexity

Ψ1ϵ(t)

p

θ le cθ 1113946t

0(t minus r)

θminus 1 ϕϵθ(r)

θdr1113888 1113889

p

le cθ suprisin[0t]

ϕϵθ(r)

p

θ 1113946t

0(t minus r)

θminus 1dr1113888 1113889

p

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus θ

minus A1( 1113857θe

c1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

p

H

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus 2θ

(r minus σ)minus 2θ

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)dσ1113874 1113875p2

+ cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus pθ

(r minus σ)minus pθ

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)dσ le cpθT sup

risin[0t]

1113946r

01113946Z

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)1113874 1113875p2

dσ1113890

times 1113946r

0(r minus σ)

minus 4pθ(pminus 2)dσ1113874 1113875(pminus 2)2

1113891 + cpθT suprisin[0t]

1113946r

0(r minus σ)

minus 2pθdσ middot supσisin[0r]

1113946Z

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)1113888 11138891113890 1113891

le cpθT 1 + supσisin[0T]

uϵ(σ)

p

H1113888 1113889

(65)

So due to (30) we get

E suptisin[0T]

Ψ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (66)

Hence if we choose θ ≔ 18and1113957θand1113954θ thanks to (60)ndash(62)and (66) for any pge 2 and θlt θ we get

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (67)

Using the Holder inequality we can estimate (58) forp 1

Lemma 3 Under (A1)ndash(A4) for any θ isin [0 θ) and 0le hle 1there exists β(θ)gt 0 such that for anyTgt 0 pge 1 x isin D((minus A1)

θ) y isin H and t isin [0 T] it holds

supϵisin(01]

E uϵ(t) minus uϵ(t + h)

p

H

le cpθT hβ(θ)p

+ h1113872 1113873 1 +xp

θ +yp

H1113872 1113873

(68)

Proof For any tge 0 with t t + h isin [0 T] we have

uϵ(t + h) minus uϵ(t) U1(t + h t) minus I( 1113857uϵ(t) + ψ1 uϵ t( 1113857(t + h) + 1113946t+h

tU1(t + h r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t+h

tU1(t + h r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t+h

t1113946Z

U1(t + h r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz) ≔ 11139445

i1I

it

(69)

By proceeding as the proof of Proposition 44 in [12] and(60) fix θ isin [0 θ) for any pge 1 and it is possible to showthat

E I1t

p

Hle cpθTh

θp 1 +xp

θ +yp

H1113872 1113873

E I2t

p

Hle cpTh

pminus 1 1 +xp

θ +yp

H1113872 1113873

E I3t

p

Hle cpTh

p 1 +xp

θ +yp

H1113872 1113873

E I4t

p

Hle cpTh

(pminus 2)2minus β1 ρ1minus 2( )ρ1( )(p2) 1 +xp

θ +yp

H1113872 1113873

(70)

According to the proof of (39) using the Holder in-equality and (30) we have

E I5t

p

Hle cpE 1113946

t+h

t1 + uϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+ cpE1113946t+h

t1 + uϵ(r)

p

H1113872 1113873dr le cp h

(pminus 2)2+ 11113872 1113873

middot 1113946t+h

t1 + E uϵ(r)

p

H1113872 1113873dr

le cpT hp

2 + h1113874 1113875 1 +xp

H +yp

H1113872 1113873

(71)

en if we take pgt 1 such that

β1 ρ1 minus 2( 1113857

ρ1

p

p minus 2lt 1 (72)

We can get

Complexity 9

E uϵ(t + h) minus uϵ(t)

p

H

le cpT h(1minus 1p )p

+ hp

+ h(12)minus (1p)minus β1 ρ1minus 2( )2ρ1( )p

+ hp2

+ h1113874 1113875

times 1 +xp

H +yp

H1113872 1113873 + cpθThθp 1 +x

p

θ +yp

H1113872 1113873

(73)

As we are assuming |h|le 1 (68) follows for any pgep bytaking

β(θ) ≔ min θ 1 minus1p

112

minus1p

minusβ1 ρ1 minus 2( 1113857

2ρ112

1113896 1113897 (74)

From the Holder inequality we can estimate (68) forpltp

E uϵ(t + h) minus uϵ(t)

p

Hle E uϵ(t + h) minus uϵ(t)

p

H1113876 1113877

pp

(75)

So we have (68)According to the above lemma we get that for everyϵ isin (0 1] the function uϵ(t) is uniformly bounded aboutt isin [0 T] and it is also equicontinuous at every point oft isin [0 T] In view of eorem 123 in [43] we can infer thatthe set uϵ1113864 1113865ϵisin(01] is relatively compact in D([0 T]H) Inaddition according to eorem 132 in [43] and the abovelemma by using Chebyshevrsquos inequality we also can get thatthe family of probability measures L(uϵ)1113864 1113865ϵisin(01] is tight inP(D([0 T]H))

4 An Evolution Family of Measures for theFast Equation

For any frozen slow component x isin H any initial conditiony isin H and any s isin R we introduce the following problem

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) v(s) y

(76)

where

wQ2(t)

wQ21 (t)

wQ22 (minus t)

⎧⎨

⎩

if tge 0

if tlt 0

1113957N2prime(t z)

1113957N1prime(t z)

1113957N3prime(minus t z)

⎧⎪⎪⎨

⎪⎪⎩

if tge 0

if tlt 0

(77)

for two independent Q2-Wiener processes wQ21 (t) and

wQ22 (t) and two independent compensated Poisson mea-

sures 1113957N1prime(dt dz) and 1113957N3prime(dt dz) with the same Levymeasure are both defined as in Section 2

According to the definition of the operator ψα2(middot s) weknow that the mapping ψα2(middot s) C([s T] H)⟶C([s T]H) is a linearly bounded operator and it is Lip-schitz continuous Hence we have that for anyx y isin H pge 1 and sltT there exists a unique mild solution

vx(middot s y) isin Lp(Ω D([0 T]H)) (see [28]) in the followingform

vx(t s y) Uα2(t s)y + ψα2 v

x(middot s y) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s y)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s y)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s y) z( 1113857 1113957N2prime(dr dz)

(78)

Moreover if the space D(RH) endowed with the to-pology of uniform convergence on bounded intervals anFt1113864 1113865tisinR-adapted process vx isin Lp(Ω D([0 T]H)) is a mildsolution of the equation

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt

+ F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz)

(79)

where t isin R en for every slt t we have

vx(t) Uα2(t s)v

x(s) + ψα2 v

x s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r) z( 1113857 1113957N2prime

(dr dz)

(80)

In the following for any x isin H and any adapted processv we set

Γα(v s)(t) ≔ 1113946t

sUα2(t r)F2(r x v(r))dw

Q2(r) tgt s

Ψα(v s)(t) ≔ 1113946t

s1113946Z

Uα2(t r)G2(r x v(r) z) 1113957N2prime(dr dz) tgt s

(81)

For any 0lt δ lt α and any v1 and v2 with slt t by pro-ceeding as in the proof of Lemma 71 in [44] it is possible toshow that there exists pgt 1 such that for any pgep we have

suprisin[st]

eδp(rminus s)

E Γα v1 s( 1113857(r) minus Γα v2 s( 1113857(r)

p

H

le cp1L

p

f2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(82)

where Lf2is the Lipschitz constant of f2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 and slt tFor the stochastic term Ψα(v s)(t) using Kunitarsquos first

inequality ([26] eorem 4423) we get

10 Complexity

E Ψα v1 s( 1113857(t) minus Ψα v2 s( 1113857(t)

p

Hle cpE 1113946

t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

p

Hv2prime(dz)dr

le cpE 1113946t

s1113946Z

eminus 2α(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

2H

v2prime(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus αp(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

p

Hv2prime(dz)dr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)e

minus 2δ(tminus s)e2δ(rminus s)

E v1(r) minus v2(r)

2Hdr1113888 1113889

p2

+ cpLpg2

1113946t

se

minus p(αminus δ)(tminus r)e

minus δp(tminus s)eδp(rminus s)

E v1(r) minus v2(r)

p

Hdr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)dr1113888 1113889

p2

+ 1113946t

se

minus p(αminus δ)(tminus r)dr⎡⎣ ⎤⎦ times eminus δp(tminus s) sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

le cp1L

pg2

(α minus δ)cp2e

minus δp(tminus s) suprisin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(83)

so

suprisin[st]

eδp(rminus s)

E Ψα v1 s( 1113857(r) minus Ψα v2 s( 1113857(r)

p

Hle cp1

Lpg2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H (84)

where Lg2is the Lipschitz constant of g2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 andslt t

Moreover using (A4) we can show that

suprisin[st]

eδp(rminus s)

E Γα(v s)(r)

p

Hle cp1

Mp

f2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(85)

suprisin[st]

eδp(rminus s)

E Ψα(v s)(r)

p

Hle cp1

Mpg2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(86)

where Mf2andMg2

are the linear growth constants off2 andg2 and cp1 and cp2 are two suitable positive constantsindependent of αgt 0 and slt t

For any fixed adapted process v let us introduce theproblem

dρ(t) A2(t) minus α( 1113857ρ(t)dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) ρ(s) 0

(87)

We denote that its unique mild solution is ρα(v s) ismeans that ρα(v s) solves the equation

ρα(v s)(t) ψα2 ρα(v s) s( 1113857(t) + Γα(v s)(t)

+ Ψα(v s)(t) slt tltT(88)

Due to (82) and (84) using the same argument as (58) in[25] it is easy to prove that for any process v1 v2 and0lt δ lt α we have

suprisin[st]

eδp(rminus s)

E ρα v1 s( 1113857(r) minus ρα v2 s( 1113857(r)

p

H

le cp1Lp

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(89)

where L max Lb2 Lf2

Lg21113966 1113967

Similarly thanks to (85) and (86) for any process v and0lt δ lt α we can prove that

suprisin[st]

eδp(rminus s)

E ρα(v s)(r)

p

Hle cp1

Mp

(α minus δ)cp2

suprisin[st]

eδp(rminus s)

E 1 +v(r)p

H1113872 1113873

(90)

where M max Mb2 Mf2

Mg21113966 1113967

Complexity 11

Lemma 4 Under (A1)ndash(A4) there exists δ gt 0 such that forany x y isin H and pge 1

E vx(t s y)

p

Hle cp 1 +x

p

H + eminus δp(tminus s)

yp

H1113872 1113873 slt t

(91)

Proof We set Λα(t) ≔ vx(t s y) minus ρα(t) whereρα(t) ρα(vx(middot s y) s)(t) is the solution of problem (87)with v vx(middot s y) Using Youngrsquos inequality we have1p

ddtΛα(t)

p

Hlelang A2(t) minus α( 1113857Λα(t)Λα(t)rangH Λα(t)

pminus 2H

+langB2 t xΛα(t) + ρα(t)( 1113857 minus B2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2H

+langB2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2Hle minus α Λα(t)

p

H

+ c Λα(t)

p

H+ c 1 +xH + ρα(t)

H1113872 1113873 Λα(t)

pminus 1H

le minus α Λα(t)

p

H+ cp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(92)

Because α is large enough we can find η α minus cp gt 0such that

ddtΛα(t)

p

Hle minus ηp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(93)

According to the Gronwall inequality we have

Λα(t)

p

Hle e

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(94)

So for any pge 1

vx(t s y)

p

Hle cp ρα(t)

p

H+ cpe

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(95)

Fixing 0lt δ lt η according to (90) we get

eδp(tminus s)

E vx(t s y)

p

Hle cpe

δp(tminus s)E ρα(t)

p

H+ cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873 + cp 1113946t

seδp(rminus s)

E ρα(r)

p

Hdr

le cp1Mp

(α minus δ)cp2sup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873 + cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ cp1Mp

(α minus δ)cp21113946

t

ssup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873dr

(96)

Taking α1 (2cp1Mp)1cp2 + δ when αge α1 we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ 1113946t

ssup

risin[st]

eδp(rminus s)

E vx(r s y)

p

Hdr

(97)

Due to the Gronwall lemma we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

(98)

Hence we get (91)

Lemma 5 Under (A1)ndash(A4) for any t isin R and x y isin H forall pge 1 there exists ηx(t) isin Lp(ΩH) such that

lims⟶minus infin

E vx(t s y) minus ηx

(t)

p

H 0 (99)

Moreover for any pge 1 there exists some δp gt 0 suchthat

E vx(t s y) minus ηx

(t)

p

Hle cpe

minus δp(tminus s) 1 +xp

H +yp

H1113872 1113873

(100)

Finally ηx is a mild solution in R of (79)

Proof Fix hgt 0 and define

ρ(t) vx(t s y) minus v

x(t s minus h y) slt t (101)

We know that ρ(t) is the unique mild solution of theproblem

12 Complexity

dρ(t) A2(t) minus α( 1113857ρ(t) + B2 t x vx(t s y)( 1113857 minus B2 t x vx(t s minus h y)( 11138571113858 1113859dt

+ F2 t x vx(t s y)( 1113857 minus F2 t x vx(t s minus h y)( 11138571113858 1113859dwQ2(t)

+1113946Z

G2 t x vx(t s y)( 1113857 minus G2 t x vx(t s minus h y)( 11138571113858 1113859 1113957N2prime(dt dz)

ρ(s) y minus vx(s s minus h y)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(t) Uα2(t s) y minus vx(s s minus h y)( 1113857 + ψα2(ρ s)(t)

+ 1113946t

sUα2(t r) B2 r x v

x(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

+ 1113946t

sUα2(t r) F2 r x v

x(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r) G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime

(dr dz)

(102)

Multiply both sides of the above equation by eδp(tminus s)Because α large enough according to Lemma 24 in [25] wehave

eδp(tminus s)

Eρ(t)p

H le cpe(δminus α)p(tminus s)

E ec2(ts)A2 y minus v

x(s s minus h y)( 1113857

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

B2 r x vx(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

p

H

+ cpE 1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime(dr dz)

p

H

≔ 11139444

i1I

it

(103)

According to Lemma 31 in [12] we know that for anyJ isinL(Linfin(D)H)capL(H L1(D)) with J Jlowast and sge 0 wehave

esAi JQi

22 leKis

minus βi ρi minus 2( )ρi eminus α ρi+2( )ρi( )s

J2L Linfin(D)H( )

(104)

where

Ki βi

e1113888 1113889

βi ρi minus 2( )ρi

ζ ρi minus 2( )ρi

i κ2ρi

i (105)

Taking pgt 1 such that pβ2(ρ2 minus 2)[ρ2(p minus 2)]lt 1en using the BurkholderndashDavisndashGundy inequality andKunitarsquos first inequality we can get that for any pgep and0lt δ lt α it yields

I2t le cpL

p

b2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se

(δminus α)(tminus r)dr1113888 1113889

p

le cp

Lp

b2

(α minus δ)p suprisin[st]

eδp(rminus s)

Eρ(r)p

H (106)

Complexity 13

I3t le cpE 1113946

t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859Q2

2

2dr1113888 1113889

p2

le cpLp

f2K

p22 1113946

t

se2(δminus α)(tminus r)

c2(t r)minus β2 ρ2minus 2( )ρ2( )e

minus α ρ2+2( )ρ2c2(tr)dr1113888 1113889

p2

times suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se2(δminus α)(tminus r)

c0(t minus r)1113858 1113859minus β2 ρ2minus 2( )ρ2( )dr1113888 1113889

p2

le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0r

minus β2 ρ2minus 2( )ρ2( )(p(pminus 2))dr1113888 1113889

(pminus 2)2

1113946tminus s

0e

minus p(αminus δ)rdr1113888 1113889

le cp

Lp

f2

α minus δsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

I4t le cpE 1113946

t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

p

Hv2prime

(dz)dr

le cpLpg2

suprisin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0e

minus 2(αminus δ)rdr1113888 1113889

p2

+ 1113946tminus s

0e

minus p(αminus δ)rdr⎡⎣ ⎤⎦le cp

Lpg2

(α minus δ)cpsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(107)

Hence we have

suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H

+ cp1Lp

(α minus δ)cp2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(108)

erefore for αgt 0 large enough we can find 0lt δp lt αsuch that

cp1L

α minus δp1113872 1113873cp2lt 1 (109)

is implies that

suprisin[st]

epδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H (110)

Let δp pδp thanks to Lemma 4 we have

E vx(t s y) minus v

x(t s minus h y)

p

H

le cpeminus δp(tminus s)

y minus vx(s s minus h y)

p

H

le cpeminus δp(tminus s) 1 +x

p

H +yp

H + eminus δph

yp

H1113872 1113873

(111)

Because Lp(ΩH) has completeness for any pgep lets⟶ minus infin in (111) there exists ηx(t) isin Lp(ΩH) such that(99) holds en if we let h⟶infin in (111) we obtain (100)Using the Holder inequality we can get (99) and (100) holdsfor any pltp

If we take y1 y2 isin H use the same arguments forvx(t s y1) minus vx(t s y2) slt t we have

E vx

t s y1( 1113857 minus vx

t s y2( 1113857

p

Hle cpe

minus δp(tminus s)y1 minus y2

p

H slt t

(112)

Let s⟶ minus infin this means that the limit ηx(t) does notdepend on the initial condition y isin H

Finally we prove that ηx(t) is a mild solution of (79)Due to the limit ηx(t) does not depend on the initialcondition we can let initial condition y 0 For any slt t

and hgt 0 we have

vx(t s minus h 0) Uα2(t s)v

x(s s minus h 0) + ψα2 v

x(middot s minus h 0) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s minus h 0)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s minus h 0)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s minus h 0) z( 1113857 1113957N2prime

(dr dz)

(113)

Let h⟶infin on both sides due to (99) we can get forany slt t have

ηx(t) Uα2(t s)ηx

(s) + ψα2 ηx s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x ηx

(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x ηx

(r)( 1113857dwQ2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x ηx(r) z( 1113857 1113957N2prime

(dr dz)

(114)

is means that ηx(t) is a mild solution of (79)For any t isin R and x isin H we denote that the law of the

random variable ηx(t) is μxt and we introduce the transition

evolution operator

Pxstφ(y) Eφ v

x(t s y)( 1113857 slt t y isin H (115)

14 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

By dealing with the Poisson terms we prove that anevolution family of measures (μx

t t isin R) on L2(O) for fastequation (4) exists en assuming thatA2(t) is periodic andb2 f2 andg2 are almost periodic we prove that the evolutionfamily ofmeasures is almost periodicMoreover with the aid ofeorem 210 in [32] we prove that the family of functions

t isin R⟼1113946L2(O)

B1(t x y)μxt (dy)1113896 1113897 (5)

is uniformly almost periodic for any x in a compact setK sub L2(O) where B1(t x y)(ξ) b1(t ξ x(ξ) y(ξ)) forany x y isin L2(O) and ξ isin O

According to the characteristics of almost periodicfunction ([25] eorem 34) we define the averaged co-efficient B1 as follows

B1(x) ≔ limT⟶infin

1T

1113946T

01113946

L2(O)B1(t x y)μx

t (dy)dt x isin L2(O)

(6)

Finally the averaged equation is obtained through theaveraged coefficient B1 Using the classical Khasminskiimethod to the present situation the averaging principle iseffective

e abovementioned notations will be given in Section 2and in this paper cgt 0 below with or without subscripts willrepresent a universal constant whose value may vary indifferent occasions

2 Notations Assumptions and Preliminaries

Let O be a bounded domain of Rd(dge 1) having a smoothboundary In this paper we denote H the separable Hilbertspace L2(O) endowed with the usual scalar product

langx yrangH 1113946O

x(ξ)y(ξ)dξ (7)

and with the corresponding norm middotH e norm in Linfin(O)

will be denoted by middotinfinFurthermore the subspace D((minus A)θ) [33ndash35] of the

generator A is dense in H and endowed with the norm

Λθ (minus A)θΛ

H Λ isin D (minus A)

θ1113872 1113873 (8)

for 0le θlt 1 0lt tleT According to eorem 613 in [36]there exists cθ gt 0 such that

(minus A)θe

At

Hle cθt

minus θ (9)

We denote by Bb(H) the Banach space of the boundedBorel functions φ H⟶ R endowed with the sup-norm

φ0 ≔ supxisinH

|φ(x)| (10)

and Cb(H) is the subspace of the uniformly continuousmappings Moreover D([s T]H) denotes the space of allcadlag path from [s T] into H

We shall denote that L(H) is the space of the boundedlinear operators in H and denote L2(H) the subspace ofHilbertndashSchmidt operators endowed with the norm

Q2 Tr QlowastQ[ ]

1113968 (11)

In slow-fast system (1) the Gaussian noises zωQ1 zt(t ξ)

and zωQ2 zt(t ξ) are assumed to be white in time andcolored in space in the case of space dimension dgt 1 fortge 0 and ξ isin O And ωQi (t ξ)(i 1 2) is the cylindricalWiener processes defined as

ωQi (t ξ) 1113944infin

k1Qiek(ξ)βk(t) i 1 2 (12)

where ek1113864 1113865kisinN is a complete orthonormal basis in Hβk(t)1113864 1113865kisinN is a sequence of mutually independent standardBrownian motion defined on the same complete stochasticbasis (ΩF Ft1113864 1113865tge0P) and Qi is a bounded linear operatoron H

Next we give the definitions of Poisson random mea-sures 1113957N1(dt dz) and 1113957N

ϵ2(dt dz) Let (ZB(Z)) be a given

measurable space and v(dz) be a σ-finite measure on itDpi

t i 1 2 represents two countable subsets of R+

Moreover let p1t t isin Dp1

tbe a stationaryFt-adapted Poisson

point process on Z with the characteristic v and p2t t isin Dp2

t

be the other stationaryFt-adapted Poisson point process onZ with the characteristic vϵ Denote by Ni(dt dz) i 1 2the Poisson counting measure associated with pi

t ie

Ni(tΛ) ≔ 1113944sisinD

pitslet

IΛ pit1113872 1113873 i 1 2

(13)

Let us denote the two independent compensated Poissonmeasures

1113957N1(dt dz) ≔ N1(dt dz) minus v1(dz)dt

1113957Nϵ2(dt dz) ≔ N2(dt dz) minus

1ϵv2(dz)dt

(14)

where v1(dz)dt and 1ϵv2(dz)dt are the compensatorsRefer to [28 37] for a more detailed description of the

stochastic integral with respect to a cylindrical Wienerprocess and Poisson random measure

For any t isin R the operators A1(t) and A2(t) are sec-ond-order uniformly elliptic operators having continuouscoefficients on O e operators N1 and N2 are theboundary operators which can be either the identity op-erator (Dirichlet boundary condition) or a first-order op-erator (coefficients satisfying a uniform nontangentialitycondition) We shall assume that the operatorAi(t) has thefollowing form

Ai(t) ci(t)Ai + Li(t) t isin R i 1 2 (15)

where Ai is a second order uniformly elliptic operator[38 39] with continuous coefficients on O which is inde-pendent of t In addition Li(t) is a first-order differentialoperator which has the form

Li(t ξ)u(ξ) langli(t ξ)nablau(ξ)rangRd t isin R ξ isin O (16)

e realizations of the differential operators Ai and Li

in H are Ai and Li Moreover A1 and A2 generate twoanalytic semigroups etA1 and etA2 respectively

Complexity 3

Now we give the following assumptions

(A1)

(a) For i 1 2 the function ci R⟶ R is con-tinuous and there exist c0 cgt 0 such that

c0 le ci(t)le c t isin R (17)

(b) For i 1 2 the function li R times O⟶ Rd iscontinuous and bounded

(A2) For i 1 2 there exists a complete orthonormalsystem eik1113966 1113967

kisinN in H and two sequences of nonneg-ative real numbers αik1113966 1113967

kisinN and λik1113966 1113967kisinN such that

Aieik minus αikeik

Qieik λikeik kge 1

κi ≔ 1113944infin

k1λρi

ik eik

2infin ltinfin

ζ i ≔ 1113944infin

k1αminus βi

ik eik

2infin ltinfin

(18)

for some constants ρi isin (2 +infin] and βi isin (0 +infin)

such that

βi ρi minus 2( 11138571113858 1113859

ρi

lt 1 (19)

Remark 1 For more comments and examples about theassumption (A2) of the operators Ai and Qi the reader canread [12]

(A3) e mappings b1 R times O times R2⟶ R f1 R times O times

R⟶ R g1 R times O times R times Z⟶ R are measur-able and the mappings b1(t ξ middot) R2⟶ R

f1(t ξ middot) R⟶ R g1(t ξ middot z) R⟶ R areLipschitz continuous and linearly growing uni-formly with respect to (t ξ z) isin R times O times Z More-over for all pge 1 there exist positive constantsc1 and c2 such that for all x1 x2 isin R we have

sup(tξ)isinRtimesO

1113946Z

g1 t ξ x1 z( 11138571113868111386811138681113868

1113868111386811138681113868pυ1(dz)

le c1 1 + x11113868111386811138681113868

1113868111386811138681113868p

1113872 1113873

sup(tξ)isinRtimesO

1113946Z

g1 t ξ x1 z( 1113857 minus g1 t ξ x2 z( 11138571113868111386811138681113868

1113868111386811138681113868pυ1(dz)

le c2 x1 minus x21113868111386811138681113868

1113868111386811138681113868p

(20)

(A4) e mappings b2 R times O times R2⟶ R f2 R times O times

R2⟶ R g2 R times O times R2 times Z⟶ R are mea-surable and the mappings b2(t ξ middot) R2⟶ R

f2(t ξ middot) R2⟶ R g2(t ξ middot z) R2⟶ R areLipschitz continuous and linearly growing uni-formly with respect to (t ξ z) isin R times O times Z

Moreover for all qge 1 there exist positive con-stants c3 and c4 such that for all(xi yi) isin R2 i 1 2 we have

sup(tξ)isinRtimesO

1113946Z

g2 t ξ x1 y1 z( 11138571113868111386811138681113868

1113868111386811138681113868qυ2(dz)

le c3 1 + x11113868111386811138681113868

1113868111386811138681113868q

+ y11113868111386811138681113868

1113868111386811138681113868q

1113872 1113873

sup(tξ)isinRtimesO

1113946Z

g2 t ξ x1 y1 z( 11138571113868111386811138681113868

minus g2 t ξ x2 y2 z( 1113857|qυ2(dz)

le c4 x1 minus x21113868111386811138681113868

1113868111386811138681113868q

+ y1 minus y21113868111386811138681113868

1113868111386811138681113868q11138731113872 (21)

Remark 2 For any (t ξ) isin R times O and x y h isin H z isin Z weshall set

B1(t x y)(ξ) ≔ b1(t ξ x(ξ) y(ξ))

B2(t x y)(ξ) ≔ b2(t ξ x(ξ) y(ξ))

F1(t x)h1113858 1113859(ξ) ≔ f1(t ξ x(ξ))h(ξ)

F2(t x y)h1113858 1113859(ξ) ≔ f2(t ξ x(ξ) y(ξ))h(ξ)

G1(t x z)h1113858 1113859(ξ) ≔ g1(t ξ x(ξ) z)h(ξ)

G2(t x y z)h1113858 1113859(ξ) ≔ g2(t ξ x(ξ) y(ξ) z)h(ξ)

(22)

Due to (A3) and (A4) for any fixed (t z) isin (RZ) themappings

B1(t middot) H times H⟶ H B2(t middot) H times H⟶ H

F1(t middot) H⟶L(H) F2(t middot) H times H⟶L(H)

G1(t middot z) H⟶L(H) G2(t middot z) H times H⟶L(H)

(23)

are Lipschitz continuous and have linear growth conditionsNow for i 1 2 we define

ci(t s) ≔ 1113946t

sci(r)dr slt t (24)

and for any ϵgt 0 and βge 0 set

Uβϵi(t s) e1ϵci(ts)Ai minus βϵ(tminus s)

slt t (25)

For ϵ 1 we write Uβi(t s) and for ϵ 1 and β 0 wewrite Ui(t s)

Next for any ϵgt 0 βge 0 and for anyu isin C([s t] W

1p0 (O)) we define

ψβϵi(u s)(r) 1ϵ

1113946r

sUβϵi(r ρ)Li(ρ)u(ρ)dρ slt rlt t

(26)

In the case ϵ 1 we write ψβi(u s)(r) and in the caseϵ 1 and β 0 we write ψi(u s)(r)

Remark 3 By using the same argument as Chapter 5 of thebook [40] we can get that there exists a unique evolution

system U(t s) eA 1113938

t

sc(r)dr

(0le sle tleT) for c(t)A More-over we also can get that ψβϵi(u s)(t) is the solution of

4 Complexity

du(t) 1ϵ

Ai(t) minus β( 1113857u(t)dt tgt s u(s) 0 (27)

3 A Priori Bounds for the Solution

With all notations introduced above system (1) can berewritten in the following abstract form

duϵ(t) A1(t)uϵ(t) + B1 t uϵ(t) vϵ(t)( 11138571113858 1113859dt

+ F1 t uϵ(t)( 1113857dωQ1(t)

+ 1113946Z

G1 t uϵ(t) z( 1113857 1113957N1(dt dz)

dvϵ(t) 1ϵ

A2(t) minus α( 1113857vϵ(t) + B2 t uϵ(t) vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ(t) vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ(t) vϵ(t) z( 1113857 1113957Nϵ2(dt dz)

uϵ(0) x vϵ(0) y

(28)

According to Remark 2 we know that the coefficients ofsystem (28) satisfy global Lipschitz and linear growthconditions and the assumptions (A1)ndash(A4) are uniformwith respect to t isin R So using the same argument as[28 41 42] it is easy to prove that for any ϵgt 0 Tgt 0 andx y isin H there exists a unique adapted mild solution (uϵ vϵ)

to system (28) in Lp(Ω D([0 T]H times H)) is means thatthere exist two unique adapted processes uϵ and vϵ inLp(Ω D([0 T]H)) such that

uϵ(t) U1(t 0)x + ψ1 uϵ 0( 1113857(t)

+ 1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

vϵ(t) Uαϵ2(t 0)y + ψαϵ2 vϵ 0( 1113857(t)

+1ϵ

1113946t

0Uαϵ2(t r)B2 r uϵ(r) vϵ(r)( 1113857dr

+1ϵ

radic 1113946t

0Uαϵ2(t r)F2 r uϵ(r) vϵ(r)( 1113857dw

Q2(r)

+ 1113946t

01113946Z

Uαϵ2(t r)G2 r uϵ(r) vϵ(r) z( 1113857 1113957Nϵ2(dr dz)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

Lemma 1 Under (A1)ndash(A4) for any pge 1 and Tgt 0 thereexists a positive constant cpT such that for any x y isin H andϵ isin (0 1] we have

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873 (30)

1113946T

0E vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (31)

Proof For fixed ϵ isin (0 1] and x y isin H for any t isin [0 T]we denote

Γ1ϵ(t) ≔ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

Ψ1ϵ(t) ≔ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

(32)

Setting Λ1ϵ(t) ≔ uϵ(t) minus Γ1ϵ(t) minus Ψ1ϵ(t) we have

ddtΛ1ϵ(t) c1(t)A1Λ1ϵ(t) + L1(t) Λ1ϵ(t) + Γ1ϵ(t)1113872

+ Ψ1ϵ(t)1113873B1 tΛ1ϵ(t) + Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873

Λ1ϵ(0) x

(33)

For any pge 2 because B1(middot) is Lipschitz continuoususing Youngrsquos inequality we have1p

ddtΛ1ϵ(t)

p

Hlangc1(t)A1Λ1ϵ(t)Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langL1(t) Λ1ϵ(t) + Γ1ϵ(t) + Ψ1ϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langB1 tΛ1ϵ(t) + Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873

minus B1 t Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langB1 t Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873

minus B1 t Ψ1ϵ(t) vϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langB1 t Ψ1ϵ(t) vϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

le c Λ1ϵ(t)

p

H+ c Γ1ϵ(t)

H Λ1ϵ(t)

pminus 1H

+ c Ψ1ϵ(t)

H Λ1ϵ(t)

pminus 1H

+ c B1 t Ψ1ϵ(t) vϵ(t)1113872 1113873

HΛ1ϵ(t)

pminus 1H

le cp Λ1ϵ(t)

p

H+ cp 1 + Γ1ϵ(t)

p

H+ Ψ1ϵ(t)

p

H+ vϵ(t)

p

H1113872 1113873

(34)

is implies that

Λ1ϵ(t)

p

Hle e

cptx

p

H + cp 1113946t

0e

cp(tminus r) 1 + Γ1ϵ(r)

p

H1113872

+ Ψ1ϵ(r)

p

H+ vϵ(r)

p

H1113873dr

(35)

Complexity 5

According to the definition of Λ1ϵ(t) for any t isin [0 T]we have

uϵ(t)

p

Hle cpT 1 +x

p

H + suprisin[0T]

Γ1ϵ(r)

p

H+ sup

risin[0T]

Ψ1ϵ(r)

p

H1113888 1113889

+ cpT 1113946T

0vϵ(r)

p

Hdr

(36)

So

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H + E suptisin[0T]

Γ1ϵ(t)

p

H1113888

+ E suptisin[0T]

Ψ1ϵ(t)

p

H1113889 + cpT 1113946

T

0E vϵ(r)

p

Hdr

(37)

According to Lemma 41 in [12] with θ 0 it is easy toprove that

E suptisin[0T]

Γ1ϵ(t)p

H le cpT 1113946T

01 + Euϵ(r)

p

H1113872 1113873dr (38)

Due to (A3) using Kunitarsquos first inequality we get

E suptisin[0T]

Ψ1ϵ(t)

p

Hle cpE 1113946

T

01113946Z

ec1(tr)A1G1 r uϵ(r) z( 1113857

2

Hv1(dz)dr1113888 1113889

p2

+ cpE1113946T

01113946Z

ec1(tr)A1G1 r uϵ(r) z( 1113857

p

Hv1(dz)dr

le cpE 1113946T

01 + uϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+ cpE1113946T

01 + uϵ(r)

p

H1113872 1113873drle cpT 1113946

T

01 + E uϵ(r)

p

H1113872 1113873dr

(39)

E suptisin[0T]

Ψ1ϵ(t)

p

Hle cpT 1113946

T

01 + E uϵ(r)

p

H1113872 1113873dr (40)

Substituting (38) and (39) into (37) we obtain

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H + 1113946T

0E vϵ(r)

p

Hdr1113888 1113889

+ cpT 1113946T

0E sup

σisin[0r]

uϵ(σ)

p

Hdr

(41)

Now we have to estimate

1113946T

0E vϵ(r)

p

Hdr (42)

For any t isin [0 T] we set

Γ2ϵ(t) ≔1ϵ

radic 1113946t

0Uαϵ2(t r)F2 r uϵ(r) vϵ(r)( 1113857dw

Q2(r)

Ψ2ϵ(t) ≔ 1113946t

01113946Z

Uαϵ2(t r)G2 r uϵ(r) vϵ(r) z( 1113857 1113957Nϵ2(dr dz)

(43)

Let Λ2ϵ(t) ≔ vϵ(t) minus Γ2ϵ(t) minus Ψ2ϵ(t) we have

ddtΛ2ϵ(t)

1ϵ

c2(t)A2 minus α( 1113857Λ2ϵ(t) +1ϵL2(t)

middot Λ2ϵ(t) + Γ2ϵ(t) + Ψ2ϵ(t)1113872 1113873

+1ϵB2 t uϵ(t)Λ2ϵ(t) + Γ2ϵ(t)1113872

+ Ψ2ϵ(t)1113873 Λ2ϵ(0) y

(44)

For any pge 2 because αgt 0 is large enough by pro-ceeding as in (34) we can get

1p

ddtΛ2ϵ(t)

p

Hle minus

α2ϵΛ2ϵ(t)

p

H

+cp

ϵ1 + uϵ(t)

p

H+ Γ2ϵ(t)

p

H+ Ψ2ϵ(t)

p

H1113872 1113873

(45)

According to the Gronwall inequality we have

Λ2ϵ(t)

p

Hle

cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r)

middot 1 + uϵ(r)

p

H+ Γ2ϵ(r)

p

H+ Ψ2ϵ(r)

p

H1113872 1113873dr + e

minus (αp2ϵ)ty

p

H

(46)

6 Complexity

According to the definition of Λ2ϵ(t) for any t isin [0 T]we have

E vϵ(t)

p

Hle cpE Γ2ϵ(t)

p

H+ cpE Ψ2ϵ(t)

p

H+ cpe

minus (αp2ϵ)ty

p

H

+cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r) 1 + E uϵ(r)

p

H+ E Γ2ϵ(r)

p

H1113872

+ E Ψ2ϵ(r)

p

H1113873dr

(47)

erefore by integrating with respect to t using Youngrsquosinequality we obtain

1113946t

0E vϵ(r)

p

Hdrle cp(t)y

p

H + cp 1113946t

0E Γ2ϵ(r)

p

Hdr + 1113946

t

0E Ψ2ϵ(r)

p

Hdr1113888 1113889 +

cp

ϵ1113946

t

01 + E uϵ(r)

p

H+ E Γ2ϵ(r)

p

H+ E Ψ2ϵ(r)

p

H1113872 1113873dr

times 1113946t

0e

minus (αp2ϵ)rdrle cp(t) 1 +yp

H1113872 1113873 + cp 1113946t

0E uϵ(r)

p

Hdr + 1113946

t

0E Γ2ϵ(r)

p

Hdr + 1113946

t

0E Ψ2ϵ(r)

p

Hdr1113888 1113889

(48)

According to the BurkholderndashDavisndashGundy inequalityby proceeding as Proposition 42 in [12] we can easily get

1113946t

0E Γ2ϵ(r)

p

Hdrle cp(t) 1113946

t

01 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(49)

Concerning the stochastic term Ψ2ϵ(t) using Kunitarsquosfirst inequality we have

E Ψ2ϵ(t)

p

Hle cpE

1ϵ

1113946t

01113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2G2 r uϵ(r) vϵ(r) z( 1113857

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

01113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2G2 r uϵ(r) vϵ(r) z( 1113857

p

Hv2(dz)dr

lecp

ϵp2E1113946

t

0e

minus (α(pminus 2)ϵ)(tminus r) 1 + uϵ(r)

p

H+ vϵ(r)

p

H1113872 1113873dr 1113946

t

0e

minus (αpϵ)(pminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵE1113946

t

0e

minus (αpϵ)(tminus r) 1 + uϵ(r)

p

H+ vϵ(r)

p

H1113872 1113873drle

cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r) 1 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(50)

By integrating with respect to t both sides and usingYoungrsquos inequality we have

1113946t

0E Ψ2ϵ(r)

p

Hdrle

cp

ϵ1113946

t

0e

minus (αp2ϵ)rdr middot 1113946t

01 + E uϵ(r)

p

H1113872

+ E vϵ(r)

p

H1113873drle cp(t) 1113946

t

01 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(51)

Substituting (49) and (51) into (48) we get

1113946t

0E vϵ(r)

p

Hdrle cp(t) 1 +y

p

H + 1113946t

0E uϵ(r)

p

Hdr1113888 1113889

+ cpT(t) 1113946t

0E vϵ(r)

p

Hdr

(52)

As cp(0) 0 and cp(t) is a continuous increasingfunction we can fix t0 gt 0 such that for any tle t0 we havecp(t)le 12 so

Complexity 7

1113946t

0E vϵ(r)

p

Hdrle cp(t) 1 +y

p

H + E suprisin[0t]

uϵ(r)

p

H1113888 1113889 t isin 0 t01113858 1113859

(53)

Due to (41) and (53) for any t isin [0 t0] we can get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT(t)E suprisin[0t]

uϵ(r)

p

H+ cpT 1113946

t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr

(54)

Similarly we also can fix 0lt t1 le t0 such that for anytle t1 we have cpT(t)le 12 so

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT 1113946t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr t isin 0 t11113858 1113859

(55)

According to the Gronwall inequality we get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(56)

Substituting (56) into (53) it yields

1113946t

0E vϵ(r)

p

Hdrle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(57)

For any pge 2 by repeating this in the intervals[t1 2t1] [2t1 3t1] etc we can easily get (31) Substituting(31) into (41) and using the Gronwall inequality again we get(30) Using the Holder inequality we can estimate (30) and(31) for p 1

Lemma 2 Under (A1)ndash(A4) there exists θgt 0 such that forany Tgt 0 pge 1 x isin D((minus A1)

θ) with θ isin [0 θ) and y isin Hthere exist a positive constant cpθT gt 0 such that

supϵisin(01]

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (58)

Proof Assuming that x isin D((minus A1)θ)(θ ge 0) for any

t isin [0 T] we have

uϵ(t) U1(t 0)x + ψ1 uϵ 0( 1113857(t)

+ 1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

(59)

Concerning the second term ψ1(uϵ 0)(t) we get

ψ1 uϵ 0( 1113857(t)

p

θ le cp 1113946t

0minus A1( 1113857

θe

c1(tr)A1L1(r)uϵ(r)dr

p

H

le cpθ 1113946t

0(t minus r)

minus θL1(r)uϵ(r)

Hdr1113888 1113889

p

le cpθ suprisin[0T]

uϵ(r)

p

H1113946

t

0(t minus r)

minus θdr1113888 1113889

p

le cpθT 1 +xp

H +yp

H1113872 1113873

(60)

For any pge 2 according to the proof of Proposition 43in [12] and thanks to (30) and (31) it is possible to show thatthere exists a 1113957θ ge 0 such that for any θle 1113957θand12 we have

E suptisin[0T]

1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

p

θle cpθT 1 +x

p

H +yp

H1113872 1113873

(61)

E suptisin[0T]

Γ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (62)

Concerning the stochastic term Ψ1ϵ(t) using the fac-torization argument we have

Ψ1ϵ(t) cθ 1113946t

0(t minus r)

θminus 1e

c1(tr)A1ϕϵθ(r)dr (63)

where

ϕϵθ(r) 1113946r

01113946Z

(r minus σ)minus θ

ec1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

(64)

Next for any pge 2 let 1113954θ ((14) minus (12p))and12p forany θle 1113954θ according to (A3) and Lemma 1 using Kunitarsquosfirst inequality and the Holder inequality we get

8 Complexity

Ψ1ϵ(t)

p

θ le cθ 1113946t

0(t minus r)

θminus 1 ϕϵθ(r)

θdr1113888 1113889

p

le cθ suprisin[0t]

ϕϵθ(r)

p

θ 1113946t

0(t minus r)

θminus 1dr1113888 1113889

p

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus θ

minus A1( 1113857θe

c1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

p

H

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus 2θ

(r minus σ)minus 2θ

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)dσ1113874 1113875p2

+ cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus pθ

(r minus σ)minus pθ

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)dσ le cpθT sup

risin[0t]

1113946r

01113946Z

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)1113874 1113875p2

dσ1113890

times 1113946r

0(r minus σ)

minus 4pθ(pminus 2)dσ1113874 1113875(pminus 2)2

1113891 + cpθT suprisin[0t]

1113946r

0(r minus σ)

minus 2pθdσ middot supσisin[0r]

1113946Z

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)1113888 11138891113890 1113891

le cpθT 1 + supσisin[0T]

uϵ(σ)

p

H1113888 1113889

(65)

So due to (30) we get

E suptisin[0T]

Ψ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (66)

Hence if we choose θ ≔ 18and1113957θand1113954θ thanks to (60)ndash(62)and (66) for any pge 2 and θlt θ we get

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (67)

Using the Holder inequality we can estimate (58) forp 1

Lemma 3 Under (A1)ndash(A4) for any θ isin [0 θ) and 0le hle 1there exists β(θ)gt 0 such that for anyTgt 0 pge 1 x isin D((minus A1)

θ) y isin H and t isin [0 T] it holds

supϵisin(01]

E uϵ(t) minus uϵ(t + h)

p

H

le cpθT hβ(θ)p

+ h1113872 1113873 1 +xp

θ +yp

H1113872 1113873

(68)

Proof For any tge 0 with t t + h isin [0 T] we have

uϵ(t + h) minus uϵ(t) U1(t + h t) minus I( 1113857uϵ(t) + ψ1 uϵ t( 1113857(t + h) + 1113946t+h

tU1(t + h r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t+h

tU1(t + h r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t+h

t1113946Z

U1(t + h r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz) ≔ 11139445

i1I

it

(69)

By proceeding as the proof of Proposition 44 in [12] and(60) fix θ isin [0 θ) for any pge 1 and it is possible to showthat

E I1t

p

Hle cpθTh

θp 1 +xp

θ +yp

H1113872 1113873

E I2t

p

Hle cpTh

pminus 1 1 +xp

θ +yp

H1113872 1113873

E I3t

p

Hle cpTh

p 1 +xp

θ +yp

H1113872 1113873

E I4t

p

Hle cpTh

(pminus 2)2minus β1 ρ1minus 2( )ρ1( )(p2) 1 +xp

θ +yp

H1113872 1113873

(70)

According to the proof of (39) using the Holder in-equality and (30) we have

E I5t

p

Hle cpE 1113946

t+h

t1 + uϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+ cpE1113946t+h

t1 + uϵ(r)

p

H1113872 1113873dr le cp h

(pminus 2)2+ 11113872 1113873

middot 1113946t+h

t1 + E uϵ(r)

p

H1113872 1113873dr

le cpT hp

2 + h1113874 1113875 1 +xp

H +yp

H1113872 1113873

(71)

en if we take pgt 1 such that

β1 ρ1 minus 2( 1113857

ρ1

p

p minus 2lt 1 (72)

We can get

Complexity 9

E uϵ(t + h) minus uϵ(t)

p

H

le cpT h(1minus 1p )p

+ hp

+ h(12)minus (1p)minus β1 ρ1minus 2( )2ρ1( )p

+ hp2

+ h1113874 1113875

times 1 +xp

H +yp

H1113872 1113873 + cpθThθp 1 +x

p

θ +yp

H1113872 1113873

(73)

As we are assuming |h|le 1 (68) follows for any pgep bytaking

β(θ) ≔ min θ 1 minus1p

112

minus1p

minusβ1 ρ1 minus 2( 1113857

2ρ112

1113896 1113897 (74)

From the Holder inequality we can estimate (68) forpltp

E uϵ(t + h) minus uϵ(t)

p

Hle E uϵ(t + h) minus uϵ(t)

p

H1113876 1113877

pp

(75)

So we have (68)According to the above lemma we get that for everyϵ isin (0 1] the function uϵ(t) is uniformly bounded aboutt isin [0 T] and it is also equicontinuous at every point oft isin [0 T] In view of eorem 123 in [43] we can infer thatthe set uϵ1113864 1113865ϵisin(01] is relatively compact in D([0 T]H) Inaddition according to eorem 132 in [43] and the abovelemma by using Chebyshevrsquos inequality we also can get thatthe family of probability measures L(uϵ)1113864 1113865ϵisin(01] is tight inP(D([0 T]H))

4 An Evolution Family of Measures for theFast Equation

For any frozen slow component x isin H any initial conditiony isin H and any s isin R we introduce the following problem

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) v(s) y

(76)

where

wQ2(t)

wQ21 (t)

wQ22 (minus t)

⎧⎨

⎩

if tge 0

if tlt 0

1113957N2prime(t z)

1113957N1prime(t z)

1113957N3prime(minus t z)

⎧⎪⎪⎨

⎪⎪⎩

if tge 0

if tlt 0

(77)

for two independent Q2-Wiener processes wQ21 (t) and

wQ22 (t) and two independent compensated Poisson mea-

sures 1113957N1prime(dt dz) and 1113957N3prime(dt dz) with the same Levymeasure are both defined as in Section 2

According to the definition of the operator ψα2(middot s) weknow that the mapping ψα2(middot s) C([s T] H)⟶C([s T]H) is a linearly bounded operator and it is Lip-schitz continuous Hence we have that for anyx y isin H pge 1 and sltT there exists a unique mild solution

vx(middot s y) isin Lp(Ω D([0 T]H)) (see [28]) in the followingform

vx(t s y) Uα2(t s)y + ψα2 v

x(middot s y) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s y)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s y)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s y) z( 1113857 1113957N2prime(dr dz)

(78)

Moreover if the space D(RH) endowed with the to-pology of uniform convergence on bounded intervals anFt1113864 1113865tisinR-adapted process vx isin Lp(Ω D([0 T]H)) is a mildsolution of the equation

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt

+ F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz)

(79)

where t isin R en for every slt t we have

vx(t) Uα2(t s)v

x(s) + ψα2 v

x s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r) z( 1113857 1113957N2prime

(dr dz)

(80)

In the following for any x isin H and any adapted processv we set

Γα(v s)(t) ≔ 1113946t

sUα2(t r)F2(r x v(r))dw

Q2(r) tgt s

Ψα(v s)(t) ≔ 1113946t

s1113946Z

Uα2(t r)G2(r x v(r) z) 1113957N2prime(dr dz) tgt s

(81)

For any 0lt δ lt α and any v1 and v2 with slt t by pro-ceeding as in the proof of Lemma 71 in [44] it is possible toshow that there exists pgt 1 such that for any pgep we have

suprisin[st]

eδp(rminus s)

E Γα v1 s( 1113857(r) minus Γα v2 s( 1113857(r)

p

H

le cp1L

p

f2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(82)

where Lf2is the Lipschitz constant of f2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 and slt tFor the stochastic term Ψα(v s)(t) using Kunitarsquos first

inequality ([26] eorem 4423) we get

10 Complexity

E Ψα v1 s( 1113857(t) minus Ψα v2 s( 1113857(t)

p

Hle cpE 1113946

t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

p

Hv2prime(dz)dr

le cpE 1113946t

s1113946Z

eminus 2α(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

2H

v2prime(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus αp(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

p

Hv2prime(dz)dr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)e

minus 2δ(tminus s)e2δ(rminus s)

E v1(r) minus v2(r)

2Hdr1113888 1113889

p2

+ cpLpg2

1113946t

se

minus p(αminus δ)(tminus r)e

minus δp(tminus s)eδp(rminus s)

E v1(r) minus v2(r)

p

Hdr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)dr1113888 1113889

p2

+ 1113946t

se

minus p(αminus δ)(tminus r)dr⎡⎣ ⎤⎦ times eminus δp(tminus s) sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

le cp1L

pg2

(α minus δ)cp2e

minus δp(tminus s) suprisin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(83)

so

suprisin[st]

eδp(rminus s)

E Ψα v1 s( 1113857(r) minus Ψα v2 s( 1113857(r)

p

Hle cp1

Lpg2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H (84)

where Lg2is the Lipschitz constant of g2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 andslt t

Moreover using (A4) we can show that

suprisin[st]

eδp(rminus s)

E Γα(v s)(r)

p

Hle cp1

Mp

f2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(85)

suprisin[st]

eδp(rminus s)

E Ψα(v s)(r)

p

Hle cp1

Mpg2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(86)

where Mf2andMg2

are the linear growth constants off2 andg2 and cp1 and cp2 are two suitable positive constantsindependent of αgt 0 and slt t

For any fixed adapted process v let us introduce theproblem

dρ(t) A2(t) minus α( 1113857ρ(t)dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) ρ(s) 0

(87)

We denote that its unique mild solution is ρα(v s) ismeans that ρα(v s) solves the equation

ρα(v s)(t) ψα2 ρα(v s) s( 1113857(t) + Γα(v s)(t)

+ Ψα(v s)(t) slt tltT(88)

Due to (82) and (84) using the same argument as (58) in[25] it is easy to prove that for any process v1 v2 and0lt δ lt α we have

suprisin[st]

eδp(rminus s)

E ρα v1 s( 1113857(r) minus ρα v2 s( 1113857(r)

p

H

le cp1Lp

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(89)

where L max Lb2 Lf2

Lg21113966 1113967

Similarly thanks to (85) and (86) for any process v and0lt δ lt α we can prove that

suprisin[st]

eδp(rminus s)

E ρα(v s)(r)

p

Hle cp1

Mp

(α minus δ)cp2

suprisin[st]

eδp(rminus s)

E 1 +v(r)p

H1113872 1113873

(90)

where M max Mb2 Mf2

Mg21113966 1113967

Complexity 11

Lemma 4 Under (A1)ndash(A4) there exists δ gt 0 such that forany x y isin H and pge 1

E vx(t s y)

p

Hle cp 1 +x

p

H + eminus δp(tminus s)

yp

H1113872 1113873 slt t

(91)

Proof We set Λα(t) ≔ vx(t s y) minus ρα(t) whereρα(t) ρα(vx(middot s y) s)(t) is the solution of problem (87)with v vx(middot s y) Using Youngrsquos inequality we have1p

ddtΛα(t)

p

Hlelang A2(t) minus α( 1113857Λα(t)Λα(t)rangH Λα(t)

pminus 2H

+langB2 t xΛα(t) + ρα(t)( 1113857 minus B2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2H

+langB2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2Hle minus α Λα(t)

p

H

+ c Λα(t)

p

H+ c 1 +xH + ρα(t)

H1113872 1113873 Λα(t)

pminus 1H

le minus α Λα(t)

p

H+ cp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(92)

Because α is large enough we can find η α minus cp gt 0such that

ddtΛα(t)

p

Hle minus ηp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(93)

According to the Gronwall inequality we have

Λα(t)

p

Hle e

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(94)

So for any pge 1

vx(t s y)

p

Hle cp ρα(t)

p

H+ cpe

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(95)

Fixing 0lt δ lt η according to (90) we get

eδp(tminus s)

E vx(t s y)

p

Hle cpe

δp(tminus s)E ρα(t)

p

H+ cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873 + cp 1113946t

seδp(rminus s)

E ρα(r)

p

Hdr

le cp1Mp

(α minus δ)cp2sup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873 + cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ cp1Mp

(α minus δ)cp21113946

t

ssup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873dr

(96)

Taking α1 (2cp1Mp)1cp2 + δ when αge α1 we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ 1113946t

ssup

risin[st]

eδp(rminus s)

E vx(r s y)

p

Hdr

(97)

Due to the Gronwall lemma we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

(98)

Hence we get (91)

Lemma 5 Under (A1)ndash(A4) for any t isin R and x y isin H forall pge 1 there exists ηx(t) isin Lp(ΩH) such that

lims⟶minus infin

E vx(t s y) minus ηx

(t)

p

H 0 (99)

Moreover for any pge 1 there exists some δp gt 0 suchthat

E vx(t s y) minus ηx

(t)

p

Hle cpe

minus δp(tminus s) 1 +xp

H +yp

H1113872 1113873

(100)

Finally ηx is a mild solution in R of (79)

Proof Fix hgt 0 and define

ρ(t) vx(t s y) minus v

x(t s minus h y) slt t (101)

We know that ρ(t) is the unique mild solution of theproblem

12 Complexity

dρ(t) A2(t) minus α( 1113857ρ(t) + B2 t x vx(t s y)( 1113857 minus B2 t x vx(t s minus h y)( 11138571113858 1113859dt

+ F2 t x vx(t s y)( 1113857 minus F2 t x vx(t s minus h y)( 11138571113858 1113859dwQ2(t)

+1113946Z

G2 t x vx(t s y)( 1113857 minus G2 t x vx(t s minus h y)( 11138571113858 1113859 1113957N2prime(dt dz)

ρ(s) y minus vx(s s minus h y)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(t) Uα2(t s) y minus vx(s s minus h y)( 1113857 + ψα2(ρ s)(t)

+ 1113946t

sUα2(t r) B2 r x v

x(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

+ 1113946t

sUα2(t r) F2 r x v

x(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r) G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime

(dr dz)

(102)

Multiply both sides of the above equation by eδp(tminus s)Because α large enough according to Lemma 24 in [25] wehave

eδp(tminus s)

Eρ(t)p

H le cpe(δminus α)p(tminus s)

E ec2(ts)A2 y minus v

x(s s minus h y)( 1113857

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

B2 r x vx(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

p

H

+ cpE 1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime(dr dz)

p

H

≔ 11139444

i1I

it

(103)

According to Lemma 31 in [12] we know that for anyJ isinL(Linfin(D)H)capL(H L1(D)) with J Jlowast and sge 0 wehave

esAi JQi

22 leKis

minus βi ρi minus 2( )ρi eminus α ρi+2( )ρi( )s

J2L Linfin(D)H( )

(104)

where

Ki βi

e1113888 1113889

βi ρi minus 2( )ρi

ζ ρi minus 2( )ρi

i κ2ρi

i (105)

Taking pgt 1 such that pβ2(ρ2 minus 2)[ρ2(p minus 2)]lt 1en using the BurkholderndashDavisndashGundy inequality andKunitarsquos first inequality we can get that for any pgep and0lt δ lt α it yields

I2t le cpL

p

b2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se

(δminus α)(tminus r)dr1113888 1113889

p

le cp

Lp

b2

(α minus δ)p suprisin[st]

eδp(rminus s)

Eρ(r)p

H (106)

Complexity 13

I3t le cpE 1113946

t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859Q2

2

2dr1113888 1113889

p2

le cpLp

f2K

p22 1113946

t

se2(δminus α)(tminus r)

c2(t r)minus β2 ρ2minus 2( )ρ2( )e

minus α ρ2+2( )ρ2c2(tr)dr1113888 1113889

p2

times suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se2(δminus α)(tminus r)

c0(t minus r)1113858 1113859minus β2 ρ2minus 2( )ρ2( )dr1113888 1113889

p2

le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0r

minus β2 ρ2minus 2( )ρ2( )(p(pminus 2))dr1113888 1113889

(pminus 2)2

1113946tminus s

0e

minus p(αminus δ)rdr1113888 1113889

le cp

Lp

f2

α minus δsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

I4t le cpE 1113946

t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

p

Hv2prime

(dz)dr

le cpLpg2

suprisin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0e

minus 2(αminus δ)rdr1113888 1113889

p2

+ 1113946tminus s

0e

minus p(αminus δ)rdr⎡⎣ ⎤⎦le cp

Lpg2

(α minus δ)cpsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(107)

Hence we have

suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H

+ cp1Lp

(α minus δ)cp2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(108)

erefore for αgt 0 large enough we can find 0lt δp lt αsuch that

cp1L

α minus δp1113872 1113873cp2lt 1 (109)

is implies that

suprisin[st]

epδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H (110)

Let δp pδp thanks to Lemma 4 we have

E vx(t s y) minus v

x(t s minus h y)

p

H

le cpeminus δp(tminus s)

y minus vx(s s minus h y)

p

H

le cpeminus δp(tminus s) 1 +x

p

H +yp

H + eminus δph

yp

H1113872 1113873

(111)

Because Lp(ΩH) has completeness for any pgep lets⟶ minus infin in (111) there exists ηx(t) isin Lp(ΩH) such that(99) holds en if we let h⟶infin in (111) we obtain (100)Using the Holder inequality we can get (99) and (100) holdsfor any pltp

If we take y1 y2 isin H use the same arguments forvx(t s y1) minus vx(t s y2) slt t we have

E vx

t s y1( 1113857 minus vx

t s y2( 1113857

p

Hle cpe

minus δp(tminus s)y1 minus y2

p

H slt t

(112)

Let s⟶ minus infin this means that the limit ηx(t) does notdepend on the initial condition y isin H

Finally we prove that ηx(t) is a mild solution of (79)Due to the limit ηx(t) does not depend on the initialcondition we can let initial condition y 0 For any slt t

and hgt 0 we have

vx(t s minus h 0) Uα2(t s)v

x(s s minus h 0) + ψα2 v

x(middot s minus h 0) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s minus h 0)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s minus h 0)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s minus h 0) z( 1113857 1113957N2prime

(dr dz)

(113)

Let h⟶infin on both sides due to (99) we can get forany slt t have

ηx(t) Uα2(t s)ηx

(s) + ψα2 ηx s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x ηx

(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x ηx

(r)( 1113857dwQ2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x ηx(r) z( 1113857 1113957N2prime

(dr dz)

(114)

is means that ηx(t) is a mild solution of (79)For any t isin R and x isin H we denote that the law of the

random variable ηx(t) is μxt and we introduce the transition

evolution operator

Pxstφ(y) Eφ v

x(t s y)( 1113857 slt t y isin H (115)

14 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

Now we give the following assumptions

(A1)

(a) For i 1 2 the function ci R⟶ R is con-tinuous and there exist c0 cgt 0 such that

c0 le ci(t)le c t isin R (17)

(b) For i 1 2 the function li R times O⟶ Rd iscontinuous and bounded

(A2) For i 1 2 there exists a complete orthonormalsystem eik1113966 1113967

kisinN in H and two sequences of nonneg-ative real numbers αik1113966 1113967

kisinN and λik1113966 1113967kisinN such that

Aieik minus αikeik

Qieik λikeik kge 1

κi ≔ 1113944infin

k1λρi

ik eik

2infin ltinfin

ζ i ≔ 1113944infin

k1αminus βi

ik eik

2infin ltinfin

(18)

for some constants ρi isin (2 +infin] and βi isin (0 +infin)

such that

βi ρi minus 2( 11138571113858 1113859

ρi

lt 1 (19)

Remark 1 For more comments and examples about theassumption (A2) of the operators Ai and Qi the reader canread [12]

(A3) e mappings b1 R times O times R2⟶ R f1 R times O times

R⟶ R g1 R times O times R times Z⟶ R are measur-able and the mappings b1(t ξ middot) R2⟶ R

f1(t ξ middot) R⟶ R g1(t ξ middot z) R⟶ R areLipschitz continuous and linearly growing uni-formly with respect to (t ξ z) isin R times O times Z More-over for all pge 1 there exist positive constantsc1 and c2 such that for all x1 x2 isin R we have

sup(tξ)isinRtimesO

1113946Z

g1 t ξ x1 z( 11138571113868111386811138681113868

1113868111386811138681113868pυ1(dz)

le c1 1 + x11113868111386811138681113868

1113868111386811138681113868p

1113872 1113873

sup(tξ)isinRtimesO

1113946Z

g1 t ξ x1 z( 1113857 minus g1 t ξ x2 z( 11138571113868111386811138681113868

1113868111386811138681113868pυ1(dz)

le c2 x1 minus x21113868111386811138681113868

1113868111386811138681113868p

(20)

(A4) e mappings b2 R times O times R2⟶ R f2 R times O times

R2⟶ R g2 R times O times R2 times Z⟶ R are mea-surable and the mappings b2(t ξ middot) R2⟶ R

f2(t ξ middot) R2⟶ R g2(t ξ middot z) R2⟶ R areLipschitz continuous and linearly growing uni-formly with respect to (t ξ z) isin R times O times Z

Moreover for all qge 1 there exist positive con-stants c3 and c4 such that for all(xi yi) isin R2 i 1 2 we have

sup(tξ)isinRtimesO

1113946Z

g2 t ξ x1 y1 z( 11138571113868111386811138681113868

1113868111386811138681113868qυ2(dz)

le c3 1 + x11113868111386811138681113868

1113868111386811138681113868q

+ y11113868111386811138681113868

1113868111386811138681113868q

1113872 1113873

sup(tξ)isinRtimesO

1113946Z

g2 t ξ x1 y1 z( 11138571113868111386811138681113868

minus g2 t ξ x2 y2 z( 1113857|qυ2(dz)

le c4 x1 minus x21113868111386811138681113868

1113868111386811138681113868q

+ y1 minus y21113868111386811138681113868

1113868111386811138681113868q11138731113872 (21)

Remark 2 For any (t ξ) isin R times O and x y h isin H z isin Z weshall set

B1(t x y)(ξ) ≔ b1(t ξ x(ξ) y(ξ))

B2(t x y)(ξ) ≔ b2(t ξ x(ξ) y(ξ))

F1(t x)h1113858 1113859(ξ) ≔ f1(t ξ x(ξ))h(ξ)

F2(t x y)h1113858 1113859(ξ) ≔ f2(t ξ x(ξ) y(ξ))h(ξ)

G1(t x z)h1113858 1113859(ξ) ≔ g1(t ξ x(ξ) z)h(ξ)

G2(t x y z)h1113858 1113859(ξ) ≔ g2(t ξ x(ξ) y(ξ) z)h(ξ)

(22)

Due to (A3) and (A4) for any fixed (t z) isin (RZ) themappings

B1(t middot) H times H⟶ H B2(t middot) H times H⟶ H

F1(t middot) H⟶L(H) F2(t middot) H times H⟶L(H)

G1(t middot z) H⟶L(H) G2(t middot z) H times H⟶L(H)

(23)

are Lipschitz continuous and have linear growth conditionsNow for i 1 2 we define

ci(t s) ≔ 1113946t

sci(r)dr slt t (24)

and for any ϵgt 0 and βge 0 set

Uβϵi(t s) e1ϵci(ts)Ai minus βϵ(tminus s)

slt t (25)

For ϵ 1 we write Uβi(t s) and for ϵ 1 and β 0 wewrite Ui(t s)

Next for any ϵgt 0 βge 0 and for anyu isin C([s t] W

1p0 (O)) we define

ψβϵi(u s)(r) 1ϵ

1113946r

sUβϵi(r ρ)Li(ρ)u(ρ)dρ slt rlt t

(26)

In the case ϵ 1 we write ψβi(u s)(r) and in the caseϵ 1 and β 0 we write ψi(u s)(r)

Remark 3 By using the same argument as Chapter 5 of thebook [40] we can get that there exists a unique evolution

system U(t s) eA 1113938

t

sc(r)dr

(0le sle tleT) for c(t)A More-over we also can get that ψβϵi(u s)(t) is the solution of

4 Complexity

du(t) 1ϵ

Ai(t) minus β( 1113857u(t)dt tgt s u(s) 0 (27)

3 A Priori Bounds for the Solution

With all notations introduced above system (1) can berewritten in the following abstract form

duϵ(t) A1(t)uϵ(t) + B1 t uϵ(t) vϵ(t)( 11138571113858 1113859dt

+ F1 t uϵ(t)( 1113857dωQ1(t)

+ 1113946Z

G1 t uϵ(t) z( 1113857 1113957N1(dt dz)

dvϵ(t) 1ϵ

A2(t) minus α( 1113857vϵ(t) + B2 t uϵ(t) vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ(t) vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ(t) vϵ(t) z( 1113857 1113957Nϵ2(dt dz)

uϵ(0) x vϵ(0) y

(28)

According to Remark 2 we know that the coefficients ofsystem (28) satisfy global Lipschitz and linear growthconditions and the assumptions (A1)ndash(A4) are uniformwith respect to t isin R So using the same argument as[28 41 42] it is easy to prove that for any ϵgt 0 Tgt 0 andx y isin H there exists a unique adapted mild solution (uϵ vϵ)

to system (28) in Lp(Ω D([0 T]H times H)) is means thatthere exist two unique adapted processes uϵ and vϵ inLp(Ω D([0 T]H)) such that

uϵ(t) U1(t 0)x + ψ1 uϵ 0( 1113857(t)

+ 1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

vϵ(t) Uαϵ2(t 0)y + ψαϵ2 vϵ 0( 1113857(t)

+1ϵ

1113946t

0Uαϵ2(t r)B2 r uϵ(r) vϵ(r)( 1113857dr

+1ϵ

radic 1113946t

0Uαϵ2(t r)F2 r uϵ(r) vϵ(r)( 1113857dw

Q2(r)

+ 1113946t

01113946Z

Uαϵ2(t r)G2 r uϵ(r) vϵ(r) z( 1113857 1113957Nϵ2(dr dz)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

Lemma 1 Under (A1)ndash(A4) for any pge 1 and Tgt 0 thereexists a positive constant cpT such that for any x y isin H andϵ isin (0 1] we have

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873 (30)

1113946T

0E vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (31)

Proof For fixed ϵ isin (0 1] and x y isin H for any t isin [0 T]we denote

Γ1ϵ(t) ≔ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

Ψ1ϵ(t) ≔ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

(32)

Setting Λ1ϵ(t) ≔ uϵ(t) minus Γ1ϵ(t) minus Ψ1ϵ(t) we have

ddtΛ1ϵ(t) c1(t)A1Λ1ϵ(t) + L1(t) Λ1ϵ(t) + Γ1ϵ(t)1113872

+ Ψ1ϵ(t)1113873B1 tΛ1ϵ(t) + Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873

Λ1ϵ(0) x

(33)

For any pge 2 because B1(middot) is Lipschitz continuoususing Youngrsquos inequality we have1p

ddtΛ1ϵ(t)

p

Hlangc1(t)A1Λ1ϵ(t)Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langL1(t) Λ1ϵ(t) + Γ1ϵ(t) + Ψ1ϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langB1 tΛ1ϵ(t) + Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873

minus B1 t Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langB1 t Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873

minus B1 t Ψ1ϵ(t) vϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langB1 t Ψ1ϵ(t) vϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

le c Λ1ϵ(t)

p

H+ c Γ1ϵ(t)

H Λ1ϵ(t)

pminus 1H

+ c Ψ1ϵ(t)

H Λ1ϵ(t)

pminus 1H

+ c B1 t Ψ1ϵ(t) vϵ(t)1113872 1113873

HΛ1ϵ(t)

pminus 1H

le cp Λ1ϵ(t)

p

H+ cp 1 + Γ1ϵ(t)

p

H+ Ψ1ϵ(t)

p

H+ vϵ(t)

p

H1113872 1113873

(34)

is implies that

Λ1ϵ(t)

p

Hle e

cptx

p

H + cp 1113946t

0e

cp(tminus r) 1 + Γ1ϵ(r)

p

H1113872

+ Ψ1ϵ(r)

p

H+ vϵ(r)

p

H1113873dr

(35)

Complexity 5

According to the definition of Λ1ϵ(t) for any t isin [0 T]we have

uϵ(t)

p

Hle cpT 1 +x

p

H + suprisin[0T]

Γ1ϵ(r)

p

H+ sup

risin[0T]

Ψ1ϵ(r)

p

H1113888 1113889

+ cpT 1113946T

0vϵ(r)

p

Hdr

(36)

So

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H + E suptisin[0T]

Γ1ϵ(t)

p

H1113888

+ E suptisin[0T]

Ψ1ϵ(t)

p

H1113889 + cpT 1113946

T

0E vϵ(r)

p

Hdr

(37)

According to Lemma 41 in [12] with θ 0 it is easy toprove that

E suptisin[0T]

Γ1ϵ(t)p

H le cpT 1113946T

01 + Euϵ(r)

p

H1113872 1113873dr (38)

Due to (A3) using Kunitarsquos first inequality we get

E suptisin[0T]

Ψ1ϵ(t)

p

Hle cpE 1113946

T

01113946Z

ec1(tr)A1G1 r uϵ(r) z( 1113857

2

Hv1(dz)dr1113888 1113889

p2

+ cpE1113946T

01113946Z

ec1(tr)A1G1 r uϵ(r) z( 1113857

p

Hv1(dz)dr

le cpE 1113946T

01 + uϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+ cpE1113946T

01 + uϵ(r)

p

H1113872 1113873drle cpT 1113946

T

01 + E uϵ(r)

p

H1113872 1113873dr

(39)

E suptisin[0T]

Ψ1ϵ(t)

p

Hle cpT 1113946

T

01 + E uϵ(r)

p

H1113872 1113873dr (40)

Substituting (38) and (39) into (37) we obtain

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H + 1113946T

0E vϵ(r)

p

Hdr1113888 1113889

+ cpT 1113946T

0E sup

σisin[0r]

uϵ(σ)

p

Hdr

(41)

Now we have to estimate

1113946T

0E vϵ(r)

p

Hdr (42)

For any t isin [0 T] we set

Γ2ϵ(t) ≔1ϵ

radic 1113946t

0Uαϵ2(t r)F2 r uϵ(r) vϵ(r)( 1113857dw

Q2(r)

Ψ2ϵ(t) ≔ 1113946t

01113946Z

Uαϵ2(t r)G2 r uϵ(r) vϵ(r) z( 1113857 1113957Nϵ2(dr dz)

(43)

Let Λ2ϵ(t) ≔ vϵ(t) minus Γ2ϵ(t) minus Ψ2ϵ(t) we have

ddtΛ2ϵ(t)

1ϵ

c2(t)A2 minus α( 1113857Λ2ϵ(t) +1ϵL2(t)

middot Λ2ϵ(t) + Γ2ϵ(t) + Ψ2ϵ(t)1113872 1113873

+1ϵB2 t uϵ(t)Λ2ϵ(t) + Γ2ϵ(t)1113872

+ Ψ2ϵ(t)1113873 Λ2ϵ(0) y

(44)

For any pge 2 because αgt 0 is large enough by pro-ceeding as in (34) we can get

1p

ddtΛ2ϵ(t)

p

Hle minus

α2ϵΛ2ϵ(t)

p

H

+cp

ϵ1 + uϵ(t)

p

H+ Γ2ϵ(t)

p

H+ Ψ2ϵ(t)

p

H1113872 1113873

(45)

According to the Gronwall inequality we have

Λ2ϵ(t)

p

Hle

cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r)

middot 1 + uϵ(r)

p

H+ Γ2ϵ(r)

p

H+ Ψ2ϵ(r)

p

H1113872 1113873dr + e

minus (αp2ϵ)ty

p

H

(46)

6 Complexity

According to the definition of Λ2ϵ(t) for any t isin [0 T]we have

E vϵ(t)

p

Hle cpE Γ2ϵ(t)

p

H+ cpE Ψ2ϵ(t)

p

H+ cpe

minus (αp2ϵ)ty

p

H

+cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r) 1 + E uϵ(r)

p

H+ E Γ2ϵ(r)

p

H1113872

+ E Ψ2ϵ(r)

p

H1113873dr

(47)

erefore by integrating with respect to t using Youngrsquosinequality we obtain

1113946t

0E vϵ(r)

p

Hdrle cp(t)y

p

H + cp 1113946t

0E Γ2ϵ(r)

p

Hdr + 1113946

t

0E Ψ2ϵ(r)

p

Hdr1113888 1113889 +

cp

ϵ1113946

t

01 + E uϵ(r)

p

H+ E Γ2ϵ(r)

p

H+ E Ψ2ϵ(r)

p

H1113872 1113873dr

times 1113946t

0e

minus (αp2ϵ)rdrle cp(t) 1 +yp

H1113872 1113873 + cp 1113946t

0E uϵ(r)

p

Hdr + 1113946

t

0E Γ2ϵ(r)

p

Hdr + 1113946

t

0E Ψ2ϵ(r)

p

Hdr1113888 1113889

(48)

According to the BurkholderndashDavisndashGundy inequalityby proceeding as Proposition 42 in [12] we can easily get

1113946t

0E Γ2ϵ(r)

p

Hdrle cp(t) 1113946

t

01 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(49)

Concerning the stochastic term Ψ2ϵ(t) using Kunitarsquosfirst inequality we have

E Ψ2ϵ(t)

p

Hle cpE

1ϵ

1113946t

01113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2G2 r uϵ(r) vϵ(r) z( 1113857

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

01113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2G2 r uϵ(r) vϵ(r) z( 1113857

p

Hv2(dz)dr

lecp

ϵp2E1113946

t

0e

minus (α(pminus 2)ϵ)(tminus r) 1 + uϵ(r)

p

H+ vϵ(r)

p

H1113872 1113873dr 1113946

t

0e

minus (αpϵ)(pminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵE1113946

t

0e

minus (αpϵ)(tminus r) 1 + uϵ(r)

p

H+ vϵ(r)

p

H1113872 1113873drle

cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r) 1 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(50)

By integrating with respect to t both sides and usingYoungrsquos inequality we have

1113946t

0E Ψ2ϵ(r)

p

Hdrle

cp

ϵ1113946

t

0e

minus (αp2ϵ)rdr middot 1113946t

01 + E uϵ(r)

p

H1113872

+ E vϵ(r)

p

H1113873drle cp(t) 1113946

t

01 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(51)

Substituting (49) and (51) into (48) we get

1113946t

0E vϵ(r)

p

Hdrle cp(t) 1 +y

p

H + 1113946t

0E uϵ(r)

p

Hdr1113888 1113889

+ cpT(t) 1113946t

0E vϵ(r)

p

Hdr

(52)

As cp(0) 0 and cp(t) is a continuous increasingfunction we can fix t0 gt 0 such that for any tle t0 we havecp(t)le 12 so

Complexity 7

1113946t

0E vϵ(r)

p

Hdrle cp(t) 1 +y

p

H + E suprisin[0t]

uϵ(r)

p

H1113888 1113889 t isin 0 t01113858 1113859

(53)

Due to (41) and (53) for any t isin [0 t0] we can get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT(t)E suprisin[0t]

uϵ(r)

p

H+ cpT 1113946

t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr

(54)

Similarly we also can fix 0lt t1 le t0 such that for anytle t1 we have cpT(t)le 12 so

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT 1113946t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr t isin 0 t11113858 1113859

(55)

According to the Gronwall inequality we get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(56)

Substituting (56) into (53) it yields

1113946t

0E vϵ(r)

p

Hdrle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(57)

For any pge 2 by repeating this in the intervals[t1 2t1] [2t1 3t1] etc we can easily get (31) Substituting(31) into (41) and using the Gronwall inequality again we get(30) Using the Holder inequality we can estimate (30) and(31) for p 1

Lemma 2 Under (A1)ndash(A4) there exists θgt 0 such that forany Tgt 0 pge 1 x isin D((minus A1)

θ) with θ isin [0 θ) and y isin Hthere exist a positive constant cpθT gt 0 such that

supϵisin(01]

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (58)

Proof Assuming that x isin D((minus A1)θ)(θ ge 0) for any

t isin [0 T] we have

uϵ(t) U1(t 0)x + ψ1 uϵ 0( 1113857(t)

+ 1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

(59)

Concerning the second term ψ1(uϵ 0)(t) we get

ψ1 uϵ 0( 1113857(t)

p

θ le cp 1113946t

0minus A1( 1113857

θe

c1(tr)A1L1(r)uϵ(r)dr

p

H

le cpθ 1113946t

0(t minus r)

minus θL1(r)uϵ(r)

Hdr1113888 1113889

p

le cpθ suprisin[0T]

uϵ(r)

p

H1113946

t

0(t minus r)

minus θdr1113888 1113889

p

le cpθT 1 +xp

H +yp

H1113872 1113873

(60)

For any pge 2 according to the proof of Proposition 43in [12] and thanks to (30) and (31) it is possible to show thatthere exists a 1113957θ ge 0 such that for any θle 1113957θand12 we have

E suptisin[0T]

1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

p

θle cpθT 1 +x

p

H +yp

H1113872 1113873

(61)

E suptisin[0T]

Γ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (62)

Concerning the stochastic term Ψ1ϵ(t) using the fac-torization argument we have

Ψ1ϵ(t) cθ 1113946t

0(t minus r)

θminus 1e

c1(tr)A1ϕϵθ(r)dr (63)

where

ϕϵθ(r) 1113946r

01113946Z

(r minus σ)minus θ

ec1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

(64)

Next for any pge 2 let 1113954θ ((14) minus (12p))and12p forany θle 1113954θ according to (A3) and Lemma 1 using Kunitarsquosfirst inequality and the Holder inequality we get

8 Complexity

Ψ1ϵ(t)

p

θ le cθ 1113946t

0(t minus r)

θminus 1 ϕϵθ(r)

θdr1113888 1113889

p

le cθ suprisin[0t]

ϕϵθ(r)

p

θ 1113946t

0(t minus r)

θminus 1dr1113888 1113889

p

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus θ

minus A1( 1113857θe

c1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

p

H

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus 2θ

(r minus σ)minus 2θ

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)dσ1113874 1113875p2

+ cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus pθ

(r minus σ)minus pθ

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)dσ le cpθT sup

risin[0t]

1113946r

01113946Z

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)1113874 1113875p2

dσ1113890

times 1113946r

0(r minus σ)

minus 4pθ(pminus 2)dσ1113874 1113875(pminus 2)2

1113891 + cpθT suprisin[0t]

1113946r

0(r minus σ)

minus 2pθdσ middot supσisin[0r]

1113946Z

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)1113888 11138891113890 1113891

le cpθT 1 + supσisin[0T]

uϵ(σ)

p

H1113888 1113889

(65)

So due to (30) we get

E suptisin[0T]

Ψ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (66)

Hence if we choose θ ≔ 18and1113957θand1113954θ thanks to (60)ndash(62)and (66) for any pge 2 and θlt θ we get

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (67)

Using the Holder inequality we can estimate (58) forp 1

Lemma 3 Under (A1)ndash(A4) for any θ isin [0 θ) and 0le hle 1there exists β(θ)gt 0 such that for anyTgt 0 pge 1 x isin D((minus A1)

θ) y isin H and t isin [0 T] it holds

supϵisin(01]

E uϵ(t) minus uϵ(t + h)

p

H

le cpθT hβ(θ)p

+ h1113872 1113873 1 +xp

θ +yp

H1113872 1113873

(68)

Proof For any tge 0 with t t + h isin [0 T] we have

uϵ(t + h) minus uϵ(t) U1(t + h t) minus I( 1113857uϵ(t) + ψ1 uϵ t( 1113857(t + h) + 1113946t+h

tU1(t + h r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t+h

tU1(t + h r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t+h

t1113946Z

U1(t + h r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz) ≔ 11139445

i1I

it

(69)

By proceeding as the proof of Proposition 44 in [12] and(60) fix θ isin [0 θ) for any pge 1 and it is possible to showthat

E I1t

p

Hle cpθTh

θp 1 +xp

θ +yp

H1113872 1113873

E I2t

p

Hle cpTh

pminus 1 1 +xp

θ +yp

H1113872 1113873

E I3t

p

Hle cpTh

p 1 +xp

θ +yp

H1113872 1113873

E I4t

p

Hle cpTh

(pminus 2)2minus β1 ρ1minus 2( )ρ1( )(p2) 1 +xp

θ +yp

H1113872 1113873

(70)

According to the proof of (39) using the Holder in-equality and (30) we have

E I5t

p

Hle cpE 1113946

t+h

t1 + uϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+ cpE1113946t+h

t1 + uϵ(r)

p

H1113872 1113873dr le cp h

(pminus 2)2+ 11113872 1113873

middot 1113946t+h

t1 + E uϵ(r)

p

H1113872 1113873dr

le cpT hp

2 + h1113874 1113875 1 +xp

H +yp

H1113872 1113873

(71)

en if we take pgt 1 such that

β1 ρ1 minus 2( 1113857

ρ1

p

p minus 2lt 1 (72)

We can get

Complexity 9

E uϵ(t + h) minus uϵ(t)

p

H

le cpT h(1minus 1p )p

+ hp

+ h(12)minus (1p)minus β1 ρ1minus 2( )2ρ1( )p

+ hp2

+ h1113874 1113875

times 1 +xp

H +yp

H1113872 1113873 + cpθThθp 1 +x

p

θ +yp

H1113872 1113873

(73)

As we are assuming |h|le 1 (68) follows for any pgep bytaking

β(θ) ≔ min θ 1 minus1p

112

minus1p

minusβ1 ρ1 minus 2( 1113857

2ρ112

1113896 1113897 (74)

From the Holder inequality we can estimate (68) forpltp

E uϵ(t + h) minus uϵ(t)

p

Hle E uϵ(t + h) minus uϵ(t)

p

H1113876 1113877

pp

(75)

So we have (68)According to the above lemma we get that for everyϵ isin (0 1] the function uϵ(t) is uniformly bounded aboutt isin [0 T] and it is also equicontinuous at every point oft isin [0 T] In view of eorem 123 in [43] we can infer thatthe set uϵ1113864 1113865ϵisin(01] is relatively compact in D([0 T]H) Inaddition according to eorem 132 in [43] and the abovelemma by using Chebyshevrsquos inequality we also can get thatthe family of probability measures L(uϵ)1113864 1113865ϵisin(01] is tight inP(D([0 T]H))

4 An Evolution Family of Measures for theFast Equation

For any frozen slow component x isin H any initial conditiony isin H and any s isin R we introduce the following problem

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) v(s) y

(76)

where

wQ2(t)

wQ21 (t)

wQ22 (minus t)

⎧⎨

⎩

if tge 0

if tlt 0

1113957N2prime(t z)

1113957N1prime(t z)

1113957N3prime(minus t z)

⎧⎪⎪⎨

⎪⎪⎩

if tge 0

if tlt 0

(77)

for two independent Q2-Wiener processes wQ21 (t) and

wQ22 (t) and two independent compensated Poisson mea-

sures 1113957N1prime(dt dz) and 1113957N3prime(dt dz) with the same Levymeasure are both defined as in Section 2

According to the definition of the operator ψα2(middot s) weknow that the mapping ψα2(middot s) C([s T] H)⟶C([s T]H) is a linearly bounded operator and it is Lip-schitz continuous Hence we have that for anyx y isin H pge 1 and sltT there exists a unique mild solution

vx(middot s y) isin Lp(Ω D([0 T]H)) (see [28]) in the followingform

vx(t s y) Uα2(t s)y + ψα2 v

x(middot s y) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s y)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s y)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s y) z( 1113857 1113957N2prime(dr dz)

(78)

Moreover if the space D(RH) endowed with the to-pology of uniform convergence on bounded intervals anFt1113864 1113865tisinR-adapted process vx isin Lp(Ω D([0 T]H)) is a mildsolution of the equation

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt

+ F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz)

(79)

where t isin R en for every slt t we have

vx(t) Uα2(t s)v

x(s) + ψα2 v

x s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r) z( 1113857 1113957N2prime

(dr dz)

(80)

In the following for any x isin H and any adapted processv we set

Γα(v s)(t) ≔ 1113946t

sUα2(t r)F2(r x v(r))dw

Q2(r) tgt s

Ψα(v s)(t) ≔ 1113946t

s1113946Z

Uα2(t r)G2(r x v(r) z) 1113957N2prime(dr dz) tgt s

(81)

For any 0lt δ lt α and any v1 and v2 with slt t by pro-ceeding as in the proof of Lemma 71 in [44] it is possible toshow that there exists pgt 1 such that for any pgep we have

suprisin[st]

eδp(rminus s)

E Γα v1 s( 1113857(r) minus Γα v2 s( 1113857(r)

p

H

le cp1L

p

f2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(82)

where Lf2is the Lipschitz constant of f2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 and slt tFor the stochastic term Ψα(v s)(t) using Kunitarsquos first

inequality ([26] eorem 4423) we get

10 Complexity

E Ψα v1 s( 1113857(t) minus Ψα v2 s( 1113857(t)

p

Hle cpE 1113946

t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

p

Hv2prime(dz)dr

le cpE 1113946t

s1113946Z

eminus 2α(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

2H

v2prime(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus αp(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

p

Hv2prime(dz)dr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)e

minus 2δ(tminus s)e2δ(rminus s)

E v1(r) minus v2(r)

2Hdr1113888 1113889

p2

+ cpLpg2

1113946t

se

minus p(αminus δ)(tminus r)e

minus δp(tminus s)eδp(rminus s)

E v1(r) minus v2(r)

p

Hdr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)dr1113888 1113889

p2

+ 1113946t

se

minus p(αminus δ)(tminus r)dr⎡⎣ ⎤⎦ times eminus δp(tminus s) sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

le cp1L

pg2

(α minus δ)cp2e

minus δp(tminus s) suprisin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(83)

so

suprisin[st]

eδp(rminus s)

E Ψα v1 s( 1113857(r) minus Ψα v2 s( 1113857(r)

p

Hle cp1

Lpg2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H (84)

where Lg2is the Lipschitz constant of g2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 andslt t

Moreover using (A4) we can show that

suprisin[st]

eδp(rminus s)

E Γα(v s)(r)

p

Hle cp1

Mp

f2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(85)

suprisin[st]

eδp(rminus s)

E Ψα(v s)(r)

p

Hle cp1

Mpg2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(86)

where Mf2andMg2

are the linear growth constants off2 andg2 and cp1 and cp2 are two suitable positive constantsindependent of αgt 0 and slt t

For any fixed adapted process v let us introduce theproblem

dρ(t) A2(t) minus α( 1113857ρ(t)dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) ρ(s) 0

(87)

We denote that its unique mild solution is ρα(v s) ismeans that ρα(v s) solves the equation

ρα(v s)(t) ψα2 ρα(v s) s( 1113857(t) + Γα(v s)(t)

+ Ψα(v s)(t) slt tltT(88)

Due to (82) and (84) using the same argument as (58) in[25] it is easy to prove that for any process v1 v2 and0lt δ lt α we have

suprisin[st]

eδp(rminus s)

E ρα v1 s( 1113857(r) minus ρα v2 s( 1113857(r)

p

H

le cp1Lp

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(89)

where L max Lb2 Lf2

Lg21113966 1113967

Similarly thanks to (85) and (86) for any process v and0lt δ lt α we can prove that

suprisin[st]

eδp(rminus s)

E ρα(v s)(r)

p

Hle cp1

Mp

(α minus δ)cp2

suprisin[st]

eδp(rminus s)

E 1 +v(r)p

H1113872 1113873

(90)

where M max Mb2 Mf2

Mg21113966 1113967

Complexity 11

Lemma 4 Under (A1)ndash(A4) there exists δ gt 0 such that forany x y isin H and pge 1

E vx(t s y)

p

Hle cp 1 +x

p

H + eminus δp(tminus s)

yp

H1113872 1113873 slt t

(91)

Proof We set Λα(t) ≔ vx(t s y) minus ρα(t) whereρα(t) ρα(vx(middot s y) s)(t) is the solution of problem (87)with v vx(middot s y) Using Youngrsquos inequality we have1p

ddtΛα(t)

p

Hlelang A2(t) minus α( 1113857Λα(t)Λα(t)rangH Λα(t)

pminus 2H

+langB2 t xΛα(t) + ρα(t)( 1113857 minus B2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2H

+langB2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2Hle minus α Λα(t)

p

H

+ c Λα(t)

p

H+ c 1 +xH + ρα(t)

H1113872 1113873 Λα(t)

pminus 1H

le minus α Λα(t)

p

H+ cp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(92)

Because α is large enough we can find η α minus cp gt 0such that

ddtΛα(t)

p

Hle minus ηp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(93)

According to the Gronwall inequality we have

Λα(t)

p

Hle e

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(94)

So for any pge 1

vx(t s y)

p

Hle cp ρα(t)

p

H+ cpe

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(95)

Fixing 0lt δ lt η according to (90) we get

eδp(tminus s)

E vx(t s y)

p

Hle cpe

δp(tminus s)E ρα(t)

p

H+ cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873 + cp 1113946t

seδp(rminus s)

E ρα(r)

p

Hdr

le cp1Mp

(α minus δ)cp2sup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873 + cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ cp1Mp

(α minus δ)cp21113946

t

ssup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873dr

(96)

Taking α1 (2cp1Mp)1cp2 + δ when αge α1 we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ 1113946t

ssup

risin[st]

eδp(rminus s)

E vx(r s y)

p

Hdr

(97)

Due to the Gronwall lemma we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

(98)

Hence we get (91)

Lemma 5 Under (A1)ndash(A4) for any t isin R and x y isin H forall pge 1 there exists ηx(t) isin Lp(ΩH) such that

lims⟶minus infin

E vx(t s y) minus ηx

(t)

p

H 0 (99)

Moreover for any pge 1 there exists some δp gt 0 suchthat

E vx(t s y) minus ηx

(t)

p

Hle cpe

minus δp(tminus s) 1 +xp

H +yp

H1113872 1113873

(100)

Finally ηx is a mild solution in R of (79)

Proof Fix hgt 0 and define

ρ(t) vx(t s y) minus v

x(t s minus h y) slt t (101)

We know that ρ(t) is the unique mild solution of theproblem

12 Complexity

dρ(t) A2(t) minus α( 1113857ρ(t) + B2 t x vx(t s y)( 1113857 minus B2 t x vx(t s minus h y)( 11138571113858 1113859dt

+ F2 t x vx(t s y)( 1113857 minus F2 t x vx(t s minus h y)( 11138571113858 1113859dwQ2(t)

+1113946Z

G2 t x vx(t s y)( 1113857 minus G2 t x vx(t s minus h y)( 11138571113858 1113859 1113957N2prime(dt dz)

ρ(s) y minus vx(s s minus h y)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(t) Uα2(t s) y minus vx(s s minus h y)( 1113857 + ψα2(ρ s)(t)

+ 1113946t

sUα2(t r) B2 r x v

x(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

+ 1113946t

sUα2(t r) F2 r x v

x(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r) G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime

(dr dz)

(102)

Multiply both sides of the above equation by eδp(tminus s)Because α large enough according to Lemma 24 in [25] wehave

eδp(tminus s)

Eρ(t)p

H le cpe(δminus α)p(tminus s)

E ec2(ts)A2 y minus v

x(s s minus h y)( 1113857

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

B2 r x vx(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

p

H

+ cpE 1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime(dr dz)

p

H

≔ 11139444

i1I

it

(103)

According to Lemma 31 in [12] we know that for anyJ isinL(Linfin(D)H)capL(H L1(D)) with J Jlowast and sge 0 wehave

esAi JQi

22 leKis

minus βi ρi minus 2( )ρi eminus α ρi+2( )ρi( )s

J2L Linfin(D)H( )

(104)

where

Ki βi

e1113888 1113889

βi ρi minus 2( )ρi

ζ ρi minus 2( )ρi

i κ2ρi

i (105)

Taking pgt 1 such that pβ2(ρ2 minus 2)[ρ2(p minus 2)]lt 1en using the BurkholderndashDavisndashGundy inequality andKunitarsquos first inequality we can get that for any pgep and0lt δ lt α it yields

I2t le cpL

p

b2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se

(δminus α)(tminus r)dr1113888 1113889

p

le cp

Lp

b2

(α minus δ)p suprisin[st]

eδp(rminus s)

Eρ(r)p

H (106)

Complexity 13

I3t le cpE 1113946

t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859Q2

2

2dr1113888 1113889

p2

le cpLp

f2K

p22 1113946

t

se2(δminus α)(tminus r)

c2(t r)minus β2 ρ2minus 2( )ρ2( )e

minus α ρ2+2( )ρ2c2(tr)dr1113888 1113889

p2

times suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se2(δminus α)(tminus r)

c0(t minus r)1113858 1113859minus β2 ρ2minus 2( )ρ2( )dr1113888 1113889

p2

le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0r

minus β2 ρ2minus 2( )ρ2( )(p(pminus 2))dr1113888 1113889

(pminus 2)2

1113946tminus s

0e

minus p(αminus δ)rdr1113888 1113889

le cp

Lp

f2

α minus δsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

I4t le cpE 1113946

t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

p

Hv2prime

(dz)dr

le cpLpg2

suprisin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0e

minus 2(αminus δ)rdr1113888 1113889

p2

+ 1113946tminus s

0e

minus p(αminus δ)rdr⎡⎣ ⎤⎦le cp

Lpg2

(α minus δ)cpsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(107)

Hence we have

suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H

+ cp1Lp

(α minus δ)cp2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(108)

erefore for αgt 0 large enough we can find 0lt δp lt αsuch that

cp1L

α minus δp1113872 1113873cp2lt 1 (109)

is implies that

suprisin[st]

epδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H (110)

Let δp pδp thanks to Lemma 4 we have

E vx(t s y) minus v

x(t s minus h y)

p

H

le cpeminus δp(tminus s)

y minus vx(s s minus h y)

p

H

le cpeminus δp(tminus s) 1 +x

p

H +yp

H + eminus δph

yp

H1113872 1113873

(111)

Because Lp(ΩH) has completeness for any pgep lets⟶ minus infin in (111) there exists ηx(t) isin Lp(ΩH) such that(99) holds en if we let h⟶infin in (111) we obtain (100)Using the Holder inequality we can get (99) and (100) holdsfor any pltp

If we take y1 y2 isin H use the same arguments forvx(t s y1) minus vx(t s y2) slt t we have

E vx

t s y1( 1113857 minus vx

t s y2( 1113857

p

Hle cpe

minus δp(tminus s)y1 minus y2

p

H slt t

(112)

Let s⟶ minus infin this means that the limit ηx(t) does notdepend on the initial condition y isin H

Finally we prove that ηx(t) is a mild solution of (79)Due to the limit ηx(t) does not depend on the initialcondition we can let initial condition y 0 For any slt t

and hgt 0 we have

vx(t s minus h 0) Uα2(t s)v

x(s s minus h 0) + ψα2 v

x(middot s minus h 0) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s minus h 0)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s minus h 0)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s minus h 0) z( 1113857 1113957N2prime

(dr dz)

(113)

Let h⟶infin on both sides due to (99) we can get forany slt t have

ηx(t) Uα2(t s)ηx

(s) + ψα2 ηx s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x ηx

(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x ηx

(r)( 1113857dwQ2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x ηx(r) z( 1113857 1113957N2prime

(dr dz)

(114)

is means that ηx(t) is a mild solution of (79)For any t isin R and x isin H we denote that the law of the

random variable ηx(t) is μxt and we introduce the transition

evolution operator

Pxstφ(y) Eφ v

x(t s y)( 1113857 slt t y isin H (115)

14 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

du(t) 1ϵ

Ai(t) minus β( 1113857u(t)dt tgt s u(s) 0 (27)

3 A Priori Bounds for the Solution

With all notations introduced above system (1) can berewritten in the following abstract form

duϵ(t) A1(t)uϵ(t) + B1 t uϵ(t) vϵ(t)( 11138571113858 1113859dt

+ F1 t uϵ(t)( 1113857dωQ1(t)

+ 1113946Z

G1 t uϵ(t) z( 1113857 1113957N1(dt dz)

dvϵ(t) 1ϵ

A2(t) minus α( 1113857vϵ(t) + B2 t uϵ(t) vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ(t) vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ(t) vϵ(t) z( 1113857 1113957Nϵ2(dt dz)

uϵ(0) x vϵ(0) y

(28)

According to Remark 2 we know that the coefficients ofsystem (28) satisfy global Lipschitz and linear growthconditions and the assumptions (A1)ndash(A4) are uniformwith respect to t isin R So using the same argument as[28 41 42] it is easy to prove that for any ϵgt 0 Tgt 0 andx y isin H there exists a unique adapted mild solution (uϵ vϵ)

to system (28) in Lp(Ω D([0 T]H times H)) is means thatthere exist two unique adapted processes uϵ and vϵ inLp(Ω D([0 T]H)) such that

uϵ(t) U1(t 0)x + ψ1 uϵ 0( 1113857(t)

+ 1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

vϵ(t) Uαϵ2(t 0)y + ψαϵ2 vϵ 0( 1113857(t)

+1ϵ

1113946t

0Uαϵ2(t r)B2 r uϵ(r) vϵ(r)( 1113857dr

+1ϵ

radic 1113946t

0Uαϵ2(t r)F2 r uϵ(r) vϵ(r)( 1113857dw

Q2(r)

+ 1113946t

01113946Z

Uαϵ2(t r)G2 r uϵ(r) vϵ(r) z( 1113857 1113957Nϵ2(dr dz)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

Lemma 1 Under (A1)ndash(A4) for any pge 1 and Tgt 0 thereexists a positive constant cpT such that for any x y isin H andϵ isin (0 1] we have

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873 (30)

1113946T

0E vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (31)

Proof For fixed ϵ isin (0 1] and x y isin H for any t isin [0 T]we denote

Γ1ϵ(t) ≔ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

Ψ1ϵ(t) ≔ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

(32)

Setting Λ1ϵ(t) ≔ uϵ(t) minus Γ1ϵ(t) minus Ψ1ϵ(t) we have

ddtΛ1ϵ(t) c1(t)A1Λ1ϵ(t) + L1(t) Λ1ϵ(t) + Γ1ϵ(t)1113872

+ Ψ1ϵ(t)1113873B1 tΛ1ϵ(t) + Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873

Λ1ϵ(0) x

(33)

For any pge 2 because B1(middot) is Lipschitz continuoususing Youngrsquos inequality we have1p

ddtΛ1ϵ(t)

p

Hlangc1(t)A1Λ1ϵ(t)Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langL1(t) Λ1ϵ(t) + Γ1ϵ(t) + Ψ1ϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langB1 tΛ1ϵ(t) + Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873

minus B1 t Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langB1 t Γ1ϵ(t) + Ψ1ϵ(t) vϵ(t)1113872 1113873

minus B1 t Ψ1ϵ(t) vϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

+langB1 t Ψ1ϵ(t) vϵ(t)1113872 1113873Λ1ϵ(t)rangH Λ1ϵ(t)

pminus 2H

le c Λ1ϵ(t)

p

H+ c Γ1ϵ(t)

H Λ1ϵ(t)

pminus 1H

+ c Ψ1ϵ(t)

H Λ1ϵ(t)

pminus 1H

+ c B1 t Ψ1ϵ(t) vϵ(t)1113872 1113873

HΛ1ϵ(t)

pminus 1H

le cp Λ1ϵ(t)

p

H+ cp 1 + Γ1ϵ(t)

p

H+ Ψ1ϵ(t)

p

H+ vϵ(t)

p

H1113872 1113873

(34)

is implies that

Λ1ϵ(t)

p

Hle e

cptx

p

H + cp 1113946t

0e

cp(tminus r) 1 + Γ1ϵ(r)

p

H1113872

+ Ψ1ϵ(r)

p

H+ vϵ(r)

p

H1113873dr

(35)

Complexity 5

According to the definition of Λ1ϵ(t) for any t isin [0 T]we have

uϵ(t)

p

Hle cpT 1 +x

p

H + suprisin[0T]

Γ1ϵ(r)

p

H+ sup

risin[0T]

Ψ1ϵ(r)

p

H1113888 1113889

+ cpT 1113946T

0vϵ(r)

p

Hdr

(36)

So

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H + E suptisin[0T]

Γ1ϵ(t)

p

H1113888

+ E suptisin[0T]

Ψ1ϵ(t)

p

H1113889 + cpT 1113946

T

0E vϵ(r)

p

Hdr

(37)

According to Lemma 41 in [12] with θ 0 it is easy toprove that

E suptisin[0T]

Γ1ϵ(t)p

H le cpT 1113946T

01 + Euϵ(r)

p

H1113872 1113873dr (38)

Due to (A3) using Kunitarsquos first inequality we get

E suptisin[0T]

Ψ1ϵ(t)

p

Hle cpE 1113946

T

01113946Z

ec1(tr)A1G1 r uϵ(r) z( 1113857

2

Hv1(dz)dr1113888 1113889

p2

+ cpE1113946T

01113946Z

ec1(tr)A1G1 r uϵ(r) z( 1113857

p

Hv1(dz)dr

le cpE 1113946T

01 + uϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+ cpE1113946T

01 + uϵ(r)

p

H1113872 1113873drle cpT 1113946

T

01 + E uϵ(r)

p

H1113872 1113873dr

(39)

E suptisin[0T]

Ψ1ϵ(t)

p

Hle cpT 1113946

T

01 + E uϵ(r)

p

H1113872 1113873dr (40)

Substituting (38) and (39) into (37) we obtain

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H + 1113946T

0E vϵ(r)

p

Hdr1113888 1113889

+ cpT 1113946T

0E sup

σisin[0r]

uϵ(σ)

p

Hdr

(41)

Now we have to estimate

1113946T

0E vϵ(r)

p

Hdr (42)

For any t isin [0 T] we set

Γ2ϵ(t) ≔1ϵ

radic 1113946t

0Uαϵ2(t r)F2 r uϵ(r) vϵ(r)( 1113857dw

Q2(r)

Ψ2ϵ(t) ≔ 1113946t

01113946Z

Uαϵ2(t r)G2 r uϵ(r) vϵ(r) z( 1113857 1113957Nϵ2(dr dz)

(43)

Let Λ2ϵ(t) ≔ vϵ(t) minus Γ2ϵ(t) minus Ψ2ϵ(t) we have

ddtΛ2ϵ(t)

1ϵ

c2(t)A2 minus α( 1113857Λ2ϵ(t) +1ϵL2(t)

middot Λ2ϵ(t) + Γ2ϵ(t) + Ψ2ϵ(t)1113872 1113873

+1ϵB2 t uϵ(t)Λ2ϵ(t) + Γ2ϵ(t)1113872

+ Ψ2ϵ(t)1113873 Λ2ϵ(0) y

(44)

For any pge 2 because αgt 0 is large enough by pro-ceeding as in (34) we can get

1p

ddtΛ2ϵ(t)

p

Hle minus

α2ϵΛ2ϵ(t)

p

H

+cp

ϵ1 + uϵ(t)

p

H+ Γ2ϵ(t)

p

H+ Ψ2ϵ(t)

p

H1113872 1113873

(45)

According to the Gronwall inequality we have

Λ2ϵ(t)

p

Hle

cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r)

middot 1 + uϵ(r)

p

H+ Γ2ϵ(r)

p

H+ Ψ2ϵ(r)

p

H1113872 1113873dr + e

minus (αp2ϵ)ty

p

H

(46)

6 Complexity

According to the definition of Λ2ϵ(t) for any t isin [0 T]we have

E vϵ(t)

p

Hle cpE Γ2ϵ(t)

p

H+ cpE Ψ2ϵ(t)

p

H+ cpe

minus (αp2ϵ)ty

p

H

+cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r) 1 + E uϵ(r)

p

H+ E Γ2ϵ(r)

p

H1113872

+ E Ψ2ϵ(r)

p

H1113873dr

(47)

erefore by integrating with respect to t using Youngrsquosinequality we obtain

1113946t

0E vϵ(r)

p

Hdrle cp(t)y

p

H + cp 1113946t

0E Γ2ϵ(r)

p

Hdr + 1113946

t

0E Ψ2ϵ(r)

p

Hdr1113888 1113889 +

cp

ϵ1113946

t

01 + E uϵ(r)

p

H+ E Γ2ϵ(r)

p

H+ E Ψ2ϵ(r)

p

H1113872 1113873dr

times 1113946t

0e

minus (αp2ϵ)rdrle cp(t) 1 +yp

H1113872 1113873 + cp 1113946t

0E uϵ(r)

p

Hdr + 1113946

t

0E Γ2ϵ(r)

p

Hdr + 1113946

t

0E Ψ2ϵ(r)

p

Hdr1113888 1113889

(48)

According to the BurkholderndashDavisndashGundy inequalityby proceeding as Proposition 42 in [12] we can easily get

1113946t

0E Γ2ϵ(r)

p

Hdrle cp(t) 1113946

t

01 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(49)

Concerning the stochastic term Ψ2ϵ(t) using Kunitarsquosfirst inequality we have

E Ψ2ϵ(t)

p

Hle cpE

1ϵ

1113946t

01113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2G2 r uϵ(r) vϵ(r) z( 1113857

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

01113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2G2 r uϵ(r) vϵ(r) z( 1113857

p

Hv2(dz)dr

lecp

ϵp2E1113946

t

0e

minus (α(pminus 2)ϵ)(tminus r) 1 + uϵ(r)

p

H+ vϵ(r)

p

H1113872 1113873dr 1113946

t

0e

minus (αpϵ)(pminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵE1113946

t

0e

minus (αpϵ)(tminus r) 1 + uϵ(r)

p

H+ vϵ(r)

p

H1113872 1113873drle

cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r) 1 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(50)

By integrating with respect to t both sides and usingYoungrsquos inequality we have

1113946t

0E Ψ2ϵ(r)

p

Hdrle

cp

ϵ1113946

t

0e

minus (αp2ϵ)rdr middot 1113946t

01 + E uϵ(r)

p

H1113872

+ E vϵ(r)

p

H1113873drle cp(t) 1113946

t

01 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(51)

Substituting (49) and (51) into (48) we get

1113946t

0E vϵ(r)

p

Hdrle cp(t) 1 +y

p

H + 1113946t

0E uϵ(r)

p

Hdr1113888 1113889

+ cpT(t) 1113946t

0E vϵ(r)

p

Hdr

(52)

As cp(0) 0 and cp(t) is a continuous increasingfunction we can fix t0 gt 0 such that for any tle t0 we havecp(t)le 12 so

Complexity 7

1113946t

0E vϵ(r)

p

Hdrle cp(t) 1 +y

p

H + E suprisin[0t]

uϵ(r)

p

H1113888 1113889 t isin 0 t01113858 1113859

(53)

Due to (41) and (53) for any t isin [0 t0] we can get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT(t)E suprisin[0t]

uϵ(r)

p

H+ cpT 1113946

t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr

(54)

Similarly we also can fix 0lt t1 le t0 such that for anytle t1 we have cpT(t)le 12 so

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT 1113946t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr t isin 0 t11113858 1113859

(55)

According to the Gronwall inequality we get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(56)

Substituting (56) into (53) it yields

1113946t

0E vϵ(r)

p

Hdrle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(57)

For any pge 2 by repeating this in the intervals[t1 2t1] [2t1 3t1] etc we can easily get (31) Substituting(31) into (41) and using the Gronwall inequality again we get(30) Using the Holder inequality we can estimate (30) and(31) for p 1

Lemma 2 Under (A1)ndash(A4) there exists θgt 0 such that forany Tgt 0 pge 1 x isin D((minus A1)

θ) with θ isin [0 θ) and y isin Hthere exist a positive constant cpθT gt 0 such that

supϵisin(01]

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (58)

Proof Assuming that x isin D((minus A1)θ)(θ ge 0) for any

t isin [0 T] we have

uϵ(t) U1(t 0)x + ψ1 uϵ 0( 1113857(t)

+ 1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

(59)

Concerning the second term ψ1(uϵ 0)(t) we get

ψ1 uϵ 0( 1113857(t)

p

θ le cp 1113946t

0minus A1( 1113857

θe

c1(tr)A1L1(r)uϵ(r)dr

p

H

le cpθ 1113946t

0(t minus r)

minus θL1(r)uϵ(r)

Hdr1113888 1113889

p

le cpθ suprisin[0T]

uϵ(r)

p

H1113946

t

0(t minus r)

minus θdr1113888 1113889

p

le cpθT 1 +xp

H +yp

H1113872 1113873

(60)

For any pge 2 according to the proof of Proposition 43in [12] and thanks to (30) and (31) it is possible to show thatthere exists a 1113957θ ge 0 such that for any θle 1113957θand12 we have

E suptisin[0T]

1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

p

θle cpθT 1 +x

p

H +yp

H1113872 1113873

(61)

E suptisin[0T]

Γ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (62)

Concerning the stochastic term Ψ1ϵ(t) using the fac-torization argument we have

Ψ1ϵ(t) cθ 1113946t

0(t minus r)

θminus 1e

c1(tr)A1ϕϵθ(r)dr (63)

where

ϕϵθ(r) 1113946r

01113946Z

(r minus σ)minus θ

ec1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

(64)

Next for any pge 2 let 1113954θ ((14) minus (12p))and12p forany θle 1113954θ according to (A3) and Lemma 1 using Kunitarsquosfirst inequality and the Holder inequality we get

8 Complexity

Ψ1ϵ(t)

p

θ le cθ 1113946t

0(t minus r)

θminus 1 ϕϵθ(r)

θdr1113888 1113889

p

le cθ suprisin[0t]

ϕϵθ(r)

p

θ 1113946t

0(t minus r)

θminus 1dr1113888 1113889

p

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus θ

minus A1( 1113857θe

c1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

p

H

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus 2θ

(r minus σ)minus 2θ

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)dσ1113874 1113875p2

+ cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus pθ

(r minus σ)minus pθ

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)dσ le cpθT sup

risin[0t]

1113946r

01113946Z

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)1113874 1113875p2

dσ1113890

times 1113946r

0(r minus σ)

minus 4pθ(pminus 2)dσ1113874 1113875(pminus 2)2

1113891 + cpθT suprisin[0t]

1113946r

0(r minus σ)

minus 2pθdσ middot supσisin[0r]

1113946Z

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)1113888 11138891113890 1113891

le cpθT 1 + supσisin[0T]

uϵ(σ)

p

H1113888 1113889

(65)

So due to (30) we get

E suptisin[0T]

Ψ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (66)

Hence if we choose θ ≔ 18and1113957θand1113954θ thanks to (60)ndash(62)and (66) for any pge 2 and θlt θ we get

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (67)

Using the Holder inequality we can estimate (58) forp 1

Lemma 3 Under (A1)ndash(A4) for any θ isin [0 θ) and 0le hle 1there exists β(θ)gt 0 such that for anyTgt 0 pge 1 x isin D((minus A1)

θ) y isin H and t isin [0 T] it holds

supϵisin(01]

E uϵ(t) minus uϵ(t + h)

p

H

le cpθT hβ(θ)p

+ h1113872 1113873 1 +xp

θ +yp

H1113872 1113873

(68)

Proof For any tge 0 with t t + h isin [0 T] we have

uϵ(t + h) minus uϵ(t) U1(t + h t) minus I( 1113857uϵ(t) + ψ1 uϵ t( 1113857(t + h) + 1113946t+h

tU1(t + h r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t+h

tU1(t + h r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t+h

t1113946Z

U1(t + h r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz) ≔ 11139445

i1I

it

(69)

By proceeding as the proof of Proposition 44 in [12] and(60) fix θ isin [0 θ) for any pge 1 and it is possible to showthat

E I1t

p

Hle cpθTh

θp 1 +xp

θ +yp

H1113872 1113873

E I2t

p

Hle cpTh

pminus 1 1 +xp

θ +yp

H1113872 1113873

E I3t

p

Hle cpTh

p 1 +xp

θ +yp

H1113872 1113873

E I4t

p

Hle cpTh

(pminus 2)2minus β1 ρ1minus 2( )ρ1( )(p2) 1 +xp

θ +yp

H1113872 1113873

(70)

According to the proof of (39) using the Holder in-equality and (30) we have

E I5t

p

Hle cpE 1113946

t+h

t1 + uϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+ cpE1113946t+h

t1 + uϵ(r)

p

H1113872 1113873dr le cp h

(pminus 2)2+ 11113872 1113873

middot 1113946t+h

t1 + E uϵ(r)

p

H1113872 1113873dr

le cpT hp

2 + h1113874 1113875 1 +xp

H +yp

H1113872 1113873

(71)

en if we take pgt 1 such that

β1 ρ1 minus 2( 1113857

ρ1

p

p minus 2lt 1 (72)

We can get

Complexity 9

E uϵ(t + h) minus uϵ(t)

p

H

le cpT h(1minus 1p )p

+ hp

+ h(12)minus (1p)minus β1 ρ1minus 2( )2ρ1( )p

+ hp2

+ h1113874 1113875

times 1 +xp

H +yp

H1113872 1113873 + cpθThθp 1 +x

p

θ +yp

H1113872 1113873

(73)

As we are assuming |h|le 1 (68) follows for any pgep bytaking

β(θ) ≔ min θ 1 minus1p

112

minus1p

minusβ1 ρ1 minus 2( 1113857

2ρ112

1113896 1113897 (74)

From the Holder inequality we can estimate (68) forpltp

E uϵ(t + h) minus uϵ(t)

p

Hle E uϵ(t + h) minus uϵ(t)

p

H1113876 1113877

pp

(75)

So we have (68)According to the above lemma we get that for everyϵ isin (0 1] the function uϵ(t) is uniformly bounded aboutt isin [0 T] and it is also equicontinuous at every point oft isin [0 T] In view of eorem 123 in [43] we can infer thatthe set uϵ1113864 1113865ϵisin(01] is relatively compact in D([0 T]H) Inaddition according to eorem 132 in [43] and the abovelemma by using Chebyshevrsquos inequality we also can get thatthe family of probability measures L(uϵ)1113864 1113865ϵisin(01] is tight inP(D([0 T]H))

4 An Evolution Family of Measures for theFast Equation

For any frozen slow component x isin H any initial conditiony isin H and any s isin R we introduce the following problem

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) v(s) y

(76)

where

wQ2(t)

wQ21 (t)

wQ22 (minus t)

⎧⎨

⎩

if tge 0

if tlt 0

1113957N2prime(t z)

1113957N1prime(t z)

1113957N3prime(minus t z)

⎧⎪⎪⎨

⎪⎪⎩

if tge 0

if tlt 0

(77)

for two independent Q2-Wiener processes wQ21 (t) and

wQ22 (t) and two independent compensated Poisson mea-

sures 1113957N1prime(dt dz) and 1113957N3prime(dt dz) with the same Levymeasure are both defined as in Section 2

According to the definition of the operator ψα2(middot s) weknow that the mapping ψα2(middot s) C([s T] H)⟶C([s T]H) is a linearly bounded operator and it is Lip-schitz continuous Hence we have that for anyx y isin H pge 1 and sltT there exists a unique mild solution

vx(middot s y) isin Lp(Ω D([0 T]H)) (see [28]) in the followingform

vx(t s y) Uα2(t s)y + ψα2 v

x(middot s y) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s y)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s y)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s y) z( 1113857 1113957N2prime(dr dz)

(78)

Moreover if the space D(RH) endowed with the to-pology of uniform convergence on bounded intervals anFt1113864 1113865tisinR-adapted process vx isin Lp(Ω D([0 T]H)) is a mildsolution of the equation

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt

+ F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz)

(79)

where t isin R en for every slt t we have

vx(t) Uα2(t s)v

x(s) + ψα2 v

x s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r) z( 1113857 1113957N2prime

(dr dz)

(80)

In the following for any x isin H and any adapted processv we set

Γα(v s)(t) ≔ 1113946t

sUα2(t r)F2(r x v(r))dw

Q2(r) tgt s

Ψα(v s)(t) ≔ 1113946t

s1113946Z

Uα2(t r)G2(r x v(r) z) 1113957N2prime(dr dz) tgt s

(81)

For any 0lt δ lt α and any v1 and v2 with slt t by pro-ceeding as in the proof of Lemma 71 in [44] it is possible toshow that there exists pgt 1 such that for any pgep we have

suprisin[st]

eδp(rminus s)

E Γα v1 s( 1113857(r) minus Γα v2 s( 1113857(r)

p

H

le cp1L

p

f2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(82)

where Lf2is the Lipschitz constant of f2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 and slt tFor the stochastic term Ψα(v s)(t) using Kunitarsquos first

inequality ([26] eorem 4423) we get

10 Complexity

E Ψα v1 s( 1113857(t) minus Ψα v2 s( 1113857(t)

p

Hle cpE 1113946

t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

p

Hv2prime(dz)dr

le cpE 1113946t

s1113946Z

eminus 2α(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

2H

v2prime(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus αp(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

p

Hv2prime(dz)dr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)e

minus 2δ(tminus s)e2δ(rminus s)

E v1(r) minus v2(r)

2Hdr1113888 1113889

p2

+ cpLpg2

1113946t

se

minus p(αminus δ)(tminus r)e

minus δp(tminus s)eδp(rminus s)

E v1(r) minus v2(r)

p

Hdr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)dr1113888 1113889

p2

+ 1113946t

se

minus p(αminus δ)(tminus r)dr⎡⎣ ⎤⎦ times eminus δp(tminus s) sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

le cp1L

pg2

(α minus δ)cp2e

minus δp(tminus s) suprisin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(83)

so

suprisin[st]

eδp(rminus s)

E Ψα v1 s( 1113857(r) minus Ψα v2 s( 1113857(r)

p

Hle cp1

Lpg2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H (84)

where Lg2is the Lipschitz constant of g2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 andslt t

Moreover using (A4) we can show that

suprisin[st]

eδp(rminus s)

E Γα(v s)(r)

p

Hle cp1

Mp

f2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(85)

suprisin[st]

eδp(rminus s)

E Ψα(v s)(r)

p

Hle cp1

Mpg2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(86)

where Mf2andMg2

are the linear growth constants off2 andg2 and cp1 and cp2 are two suitable positive constantsindependent of αgt 0 and slt t

For any fixed adapted process v let us introduce theproblem

dρ(t) A2(t) minus α( 1113857ρ(t)dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) ρ(s) 0

(87)

We denote that its unique mild solution is ρα(v s) ismeans that ρα(v s) solves the equation

ρα(v s)(t) ψα2 ρα(v s) s( 1113857(t) + Γα(v s)(t)

+ Ψα(v s)(t) slt tltT(88)

Due to (82) and (84) using the same argument as (58) in[25] it is easy to prove that for any process v1 v2 and0lt δ lt α we have

suprisin[st]

eδp(rminus s)

E ρα v1 s( 1113857(r) minus ρα v2 s( 1113857(r)

p

H

le cp1Lp

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(89)

where L max Lb2 Lf2

Lg21113966 1113967

Similarly thanks to (85) and (86) for any process v and0lt δ lt α we can prove that

suprisin[st]

eδp(rminus s)

E ρα(v s)(r)

p

Hle cp1

Mp

(α minus δ)cp2

suprisin[st]

eδp(rminus s)

E 1 +v(r)p

H1113872 1113873

(90)

where M max Mb2 Mf2

Mg21113966 1113967

Complexity 11

Lemma 4 Under (A1)ndash(A4) there exists δ gt 0 such that forany x y isin H and pge 1

E vx(t s y)

p

Hle cp 1 +x

p

H + eminus δp(tminus s)

yp

H1113872 1113873 slt t

(91)

Proof We set Λα(t) ≔ vx(t s y) minus ρα(t) whereρα(t) ρα(vx(middot s y) s)(t) is the solution of problem (87)with v vx(middot s y) Using Youngrsquos inequality we have1p

ddtΛα(t)

p

Hlelang A2(t) minus α( 1113857Λα(t)Λα(t)rangH Λα(t)

pminus 2H

+langB2 t xΛα(t) + ρα(t)( 1113857 minus B2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2H

+langB2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2Hle minus α Λα(t)

p

H

+ c Λα(t)

p

H+ c 1 +xH + ρα(t)

H1113872 1113873 Λα(t)

pminus 1H

le minus α Λα(t)

p

H+ cp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(92)

Because α is large enough we can find η α minus cp gt 0such that

ddtΛα(t)

p

Hle minus ηp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(93)

According to the Gronwall inequality we have

Λα(t)

p

Hle e

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(94)

So for any pge 1

vx(t s y)

p

Hle cp ρα(t)

p

H+ cpe

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(95)

Fixing 0lt δ lt η according to (90) we get

eδp(tminus s)

E vx(t s y)

p

Hle cpe

δp(tminus s)E ρα(t)

p

H+ cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873 + cp 1113946t

seδp(rminus s)

E ρα(r)

p

Hdr

le cp1Mp

(α minus δ)cp2sup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873 + cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ cp1Mp

(α minus δ)cp21113946

t

ssup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873dr

(96)

Taking α1 (2cp1Mp)1cp2 + δ when αge α1 we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ 1113946t

ssup

risin[st]

eδp(rminus s)

E vx(r s y)

p

Hdr

(97)

Due to the Gronwall lemma we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

(98)

Hence we get (91)

Lemma 5 Under (A1)ndash(A4) for any t isin R and x y isin H forall pge 1 there exists ηx(t) isin Lp(ΩH) such that

lims⟶minus infin

E vx(t s y) minus ηx

(t)

p

H 0 (99)

Moreover for any pge 1 there exists some δp gt 0 suchthat

E vx(t s y) minus ηx

(t)

p

Hle cpe

minus δp(tminus s) 1 +xp

H +yp

H1113872 1113873

(100)

Finally ηx is a mild solution in R of (79)

Proof Fix hgt 0 and define

ρ(t) vx(t s y) minus v

x(t s minus h y) slt t (101)

We know that ρ(t) is the unique mild solution of theproblem

12 Complexity

dρ(t) A2(t) minus α( 1113857ρ(t) + B2 t x vx(t s y)( 1113857 minus B2 t x vx(t s minus h y)( 11138571113858 1113859dt

+ F2 t x vx(t s y)( 1113857 minus F2 t x vx(t s minus h y)( 11138571113858 1113859dwQ2(t)

+1113946Z

G2 t x vx(t s y)( 1113857 minus G2 t x vx(t s minus h y)( 11138571113858 1113859 1113957N2prime(dt dz)

ρ(s) y minus vx(s s minus h y)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(t) Uα2(t s) y minus vx(s s minus h y)( 1113857 + ψα2(ρ s)(t)

+ 1113946t

sUα2(t r) B2 r x v

x(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

+ 1113946t

sUα2(t r) F2 r x v

x(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r) G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime

(dr dz)

(102)

Multiply both sides of the above equation by eδp(tminus s)Because α large enough according to Lemma 24 in [25] wehave

eδp(tminus s)

Eρ(t)p

H le cpe(δminus α)p(tminus s)

E ec2(ts)A2 y minus v

x(s s minus h y)( 1113857

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

B2 r x vx(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

p

H

+ cpE 1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime(dr dz)

p

H

≔ 11139444

i1I

it

(103)

According to Lemma 31 in [12] we know that for anyJ isinL(Linfin(D)H)capL(H L1(D)) with J Jlowast and sge 0 wehave

esAi JQi

22 leKis

minus βi ρi minus 2( )ρi eminus α ρi+2( )ρi( )s

J2L Linfin(D)H( )

(104)

where

Ki βi

e1113888 1113889

βi ρi minus 2( )ρi

ζ ρi minus 2( )ρi

i κ2ρi

i (105)

Taking pgt 1 such that pβ2(ρ2 minus 2)[ρ2(p minus 2)]lt 1en using the BurkholderndashDavisndashGundy inequality andKunitarsquos first inequality we can get that for any pgep and0lt δ lt α it yields

I2t le cpL

p

b2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se

(δminus α)(tminus r)dr1113888 1113889

p

le cp

Lp

b2

(α minus δ)p suprisin[st]

eδp(rminus s)

Eρ(r)p

H (106)

Complexity 13

I3t le cpE 1113946

t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859Q2

2

2dr1113888 1113889

p2

le cpLp

f2K

p22 1113946

t

se2(δminus α)(tminus r)

c2(t r)minus β2 ρ2minus 2( )ρ2( )e

minus α ρ2+2( )ρ2c2(tr)dr1113888 1113889

p2

times suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se2(δminus α)(tminus r)

c0(t minus r)1113858 1113859minus β2 ρ2minus 2( )ρ2( )dr1113888 1113889

p2

le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0r

minus β2 ρ2minus 2( )ρ2( )(p(pminus 2))dr1113888 1113889

(pminus 2)2

1113946tminus s

0e

minus p(αminus δ)rdr1113888 1113889

le cp

Lp

f2

α minus δsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

I4t le cpE 1113946

t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

p

Hv2prime

(dz)dr

le cpLpg2

suprisin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0e

minus 2(αminus δ)rdr1113888 1113889

p2

+ 1113946tminus s

0e

minus p(αminus δ)rdr⎡⎣ ⎤⎦le cp

Lpg2

(α minus δ)cpsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(107)

Hence we have

suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H

+ cp1Lp

(α minus δ)cp2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(108)

erefore for αgt 0 large enough we can find 0lt δp lt αsuch that

cp1L

α minus δp1113872 1113873cp2lt 1 (109)

is implies that

suprisin[st]

epδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H (110)

Let δp pδp thanks to Lemma 4 we have

E vx(t s y) minus v

x(t s minus h y)

p

H

le cpeminus δp(tminus s)

y minus vx(s s minus h y)

p

H

le cpeminus δp(tminus s) 1 +x

p

H +yp

H + eminus δph

yp

H1113872 1113873

(111)

Because Lp(ΩH) has completeness for any pgep lets⟶ minus infin in (111) there exists ηx(t) isin Lp(ΩH) such that(99) holds en if we let h⟶infin in (111) we obtain (100)Using the Holder inequality we can get (99) and (100) holdsfor any pltp

If we take y1 y2 isin H use the same arguments forvx(t s y1) minus vx(t s y2) slt t we have

E vx

t s y1( 1113857 minus vx

t s y2( 1113857

p

Hle cpe

minus δp(tminus s)y1 minus y2

p

H slt t

(112)

Let s⟶ minus infin this means that the limit ηx(t) does notdepend on the initial condition y isin H

Finally we prove that ηx(t) is a mild solution of (79)Due to the limit ηx(t) does not depend on the initialcondition we can let initial condition y 0 For any slt t

and hgt 0 we have

vx(t s minus h 0) Uα2(t s)v

x(s s minus h 0) + ψα2 v

x(middot s minus h 0) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s minus h 0)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s minus h 0)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s minus h 0) z( 1113857 1113957N2prime

(dr dz)

(113)

Let h⟶infin on both sides due to (99) we can get forany slt t have

ηx(t) Uα2(t s)ηx

(s) + ψα2 ηx s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x ηx

(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x ηx

(r)( 1113857dwQ2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x ηx(r) z( 1113857 1113957N2prime

(dr dz)

(114)

is means that ηx(t) is a mild solution of (79)For any t isin R and x isin H we denote that the law of the

random variable ηx(t) is μxt and we introduce the transition

evolution operator

Pxstφ(y) Eφ v

x(t s y)( 1113857 slt t y isin H (115)

14 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

According to the definition of Λ1ϵ(t) for any t isin [0 T]we have

uϵ(t)

p

Hle cpT 1 +x

p

H + suprisin[0T]

Γ1ϵ(r)

p

H+ sup

risin[0T]

Ψ1ϵ(r)

p

H1113888 1113889

+ cpT 1113946T

0vϵ(r)

p

Hdr

(36)

So

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H + E suptisin[0T]

Γ1ϵ(t)

p

H1113888

+ E suptisin[0T]

Ψ1ϵ(t)

p

H1113889 + cpT 1113946

T

0E vϵ(r)

p

Hdr

(37)

According to Lemma 41 in [12] with θ 0 it is easy toprove that

E suptisin[0T]

Γ1ϵ(t)p

H le cpT 1113946T

01 + Euϵ(r)

p

H1113872 1113873dr (38)

Due to (A3) using Kunitarsquos first inequality we get

E suptisin[0T]

Ψ1ϵ(t)

p

Hle cpE 1113946

T

01113946Z

ec1(tr)A1G1 r uϵ(r) z( 1113857

2

Hv1(dz)dr1113888 1113889

p2

+ cpE1113946T

01113946Z

ec1(tr)A1G1 r uϵ(r) z( 1113857

p

Hv1(dz)dr

le cpE 1113946T

01 + uϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+ cpE1113946T

01 + uϵ(r)

p

H1113872 1113873drle cpT 1113946

T

01 + E uϵ(r)

p

H1113872 1113873dr

(39)

E suptisin[0T]

Ψ1ϵ(t)

p

Hle cpT 1113946

T

01 + E uϵ(r)

p

H1113872 1113873dr (40)

Substituting (38) and (39) into (37) we obtain

E suptisin[0T]

uϵ(t)

p

Hle cpT 1 +x

p

H + 1113946T

0E vϵ(r)

p

Hdr1113888 1113889

+ cpT 1113946T

0E sup

σisin[0r]

uϵ(σ)

p

Hdr

(41)

Now we have to estimate

1113946T

0E vϵ(r)

p

Hdr (42)

For any t isin [0 T] we set

Γ2ϵ(t) ≔1ϵ

radic 1113946t

0Uαϵ2(t r)F2 r uϵ(r) vϵ(r)( 1113857dw

Q2(r)

Ψ2ϵ(t) ≔ 1113946t

01113946Z

Uαϵ2(t r)G2 r uϵ(r) vϵ(r) z( 1113857 1113957Nϵ2(dr dz)

(43)

Let Λ2ϵ(t) ≔ vϵ(t) minus Γ2ϵ(t) minus Ψ2ϵ(t) we have

ddtΛ2ϵ(t)

1ϵ

c2(t)A2 minus α( 1113857Λ2ϵ(t) +1ϵL2(t)

middot Λ2ϵ(t) + Γ2ϵ(t) + Ψ2ϵ(t)1113872 1113873

+1ϵB2 t uϵ(t)Λ2ϵ(t) + Γ2ϵ(t)1113872

+ Ψ2ϵ(t)1113873 Λ2ϵ(0) y

(44)

For any pge 2 because αgt 0 is large enough by pro-ceeding as in (34) we can get

1p

ddtΛ2ϵ(t)

p

Hle minus

α2ϵΛ2ϵ(t)

p

H

+cp

ϵ1 + uϵ(t)

p

H+ Γ2ϵ(t)

p

H+ Ψ2ϵ(t)

p

H1113872 1113873

(45)

According to the Gronwall inequality we have

Λ2ϵ(t)

p

Hle

cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r)

middot 1 + uϵ(r)

p

H+ Γ2ϵ(r)

p

H+ Ψ2ϵ(r)

p

H1113872 1113873dr + e

minus (αp2ϵ)ty

p

H

(46)

6 Complexity

According to the definition of Λ2ϵ(t) for any t isin [0 T]we have

E vϵ(t)

p

Hle cpE Γ2ϵ(t)

p

H+ cpE Ψ2ϵ(t)

p

H+ cpe

minus (αp2ϵ)ty

p

H

+cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r) 1 + E uϵ(r)

p

H+ E Γ2ϵ(r)

p

H1113872

+ E Ψ2ϵ(r)

p

H1113873dr

(47)

erefore by integrating with respect to t using Youngrsquosinequality we obtain

1113946t

0E vϵ(r)

p

Hdrle cp(t)y

p

H + cp 1113946t

0E Γ2ϵ(r)

p

Hdr + 1113946

t

0E Ψ2ϵ(r)

p

Hdr1113888 1113889 +

cp

ϵ1113946

t

01 + E uϵ(r)

p

H+ E Γ2ϵ(r)

p

H+ E Ψ2ϵ(r)

p

H1113872 1113873dr

times 1113946t

0e

minus (αp2ϵ)rdrle cp(t) 1 +yp

H1113872 1113873 + cp 1113946t

0E uϵ(r)

p

Hdr + 1113946

t

0E Γ2ϵ(r)

p

Hdr + 1113946

t

0E Ψ2ϵ(r)

p

Hdr1113888 1113889

(48)

According to the BurkholderndashDavisndashGundy inequalityby proceeding as Proposition 42 in [12] we can easily get

1113946t

0E Γ2ϵ(r)

p

Hdrle cp(t) 1113946

t

01 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(49)

Concerning the stochastic term Ψ2ϵ(t) using Kunitarsquosfirst inequality we have

E Ψ2ϵ(t)

p

Hle cpE

1ϵ

1113946t

01113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2G2 r uϵ(r) vϵ(r) z( 1113857

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

01113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2G2 r uϵ(r) vϵ(r) z( 1113857

p

Hv2(dz)dr

lecp

ϵp2E1113946

t

0e

minus (α(pminus 2)ϵ)(tminus r) 1 + uϵ(r)

p

H+ vϵ(r)

p

H1113872 1113873dr 1113946

t

0e

minus (αpϵ)(pminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵE1113946

t

0e

minus (αpϵ)(tminus r) 1 + uϵ(r)

p

H+ vϵ(r)

p

H1113872 1113873drle

cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r) 1 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(50)

By integrating with respect to t both sides and usingYoungrsquos inequality we have

1113946t

0E Ψ2ϵ(r)

p

Hdrle

cp

ϵ1113946

t

0e

minus (αp2ϵ)rdr middot 1113946t

01 + E uϵ(r)

p

H1113872

+ E vϵ(r)

p

H1113873drle cp(t) 1113946

t

01 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(51)

Substituting (49) and (51) into (48) we get

1113946t

0E vϵ(r)

p

Hdrle cp(t) 1 +y

p

H + 1113946t

0E uϵ(r)

p

Hdr1113888 1113889

+ cpT(t) 1113946t

0E vϵ(r)

p

Hdr

(52)

As cp(0) 0 and cp(t) is a continuous increasingfunction we can fix t0 gt 0 such that for any tle t0 we havecp(t)le 12 so

Complexity 7

1113946t

0E vϵ(r)

p

Hdrle cp(t) 1 +y

p

H + E suprisin[0t]

uϵ(r)

p

H1113888 1113889 t isin 0 t01113858 1113859

(53)

Due to (41) and (53) for any t isin [0 t0] we can get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT(t)E suprisin[0t]

uϵ(r)

p

H+ cpT 1113946

t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr

(54)

Similarly we also can fix 0lt t1 le t0 such that for anytle t1 we have cpT(t)le 12 so

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT 1113946t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr t isin 0 t11113858 1113859

(55)

According to the Gronwall inequality we get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(56)

Substituting (56) into (53) it yields

1113946t

0E vϵ(r)

p

Hdrle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(57)

For any pge 2 by repeating this in the intervals[t1 2t1] [2t1 3t1] etc we can easily get (31) Substituting(31) into (41) and using the Gronwall inequality again we get(30) Using the Holder inequality we can estimate (30) and(31) for p 1

Lemma 2 Under (A1)ndash(A4) there exists θgt 0 such that forany Tgt 0 pge 1 x isin D((minus A1)

θ) with θ isin [0 θ) and y isin Hthere exist a positive constant cpθT gt 0 such that

supϵisin(01]

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (58)

Proof Assuming that x isin D((minus A1)θ)(θ ge 0) for any

t isin [0 T] we have

uϵ(t) U1(t 0)x + ψ1 uϵ 0( 1113857(t)

+ 1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

(59)

Concerning the second term ψ1(uϵ 0)(t) we get

ψ1 uϵ 0( 1113857(t)

p

θ le cp 1113946t

0minus A1( 1113857

θe

c1(tr)A1L1(r)uϵ(r)dr

p

H

le cpθ 1113946t

0(t minus r)

minus θL1(r)uϵ(r)

Hdr1113888 1113889

p

le cpθ suprisin[0T]

uϵ(r)

p

H1113946

t

0(t minus r)

minus θdr1113888 1113889

p

le cpθT 1 +xp

H +yp

H1113872 1113873

(60)

For any pge 2 according to the proof of Proposition 43in [12] and thanks to (30) and (31) it is possible to show thatthere exists a 1113957θ ge 0 such that for any θle 1113957θand12 we have

E suptisin[0T]

1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

p

θle cpθT 1 +x

p

H +yp

H1113872 1113873

(61)

E suptisin[0T]

Γ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (62)

Concerning the stochastic term Ψ1ϵ(t) using the fac-torization argument we have

Ψ1ϵ(t) cθ 1113946t

0(t minus r)

θminus 1e

c1(tr)A1ϕϵθ(r)dr (63)

where

ϕϵθ(r) 1113946r

01113946Z

(r minus σ)minus θ

ec1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

(64)

Next for any pge 2 let 1113954θ ((14) minus (12p))and12p forany θle 1113954θ according to (A3) and Lemma 1 using Kunitarsquosfirst inequality and the Holder inequality we get

8 Complexity

Ψ1ϵ(t)

p

θ le cθ 1113946t

0(t minus r)

θminus 1 ϕϵθ(r)

θdr1113888 1113889

p

le cθ suprisin[0t]

ϕϵθ(r)

p

θ 1113946t

0(t minus r)

θminus 1dr1113888 1113889

p

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus θ

minus A1( 1113857θe

c1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

p

H

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus 2θ

(r minus σ)minus 2θ

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)dσ1113874 1113875p2

+ cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus pθ

(r minus σ)minus pθ

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)dσ le cpθT sup

risin[0t]

1113946r

01113946Z

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)1113874 1113875p2

dσ1113890

times 1113946r

0(r minus σ)

minus 4pθ(pminus 2)dσ1113874 1113875(pminus 2)2

1113891 + cpθT suprisin[0t]

1113946r

0(r minus σ)

minus 2pθdσ middot supσisin[0r]

1113946Z

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)1113888 11138891113890 1113891

le cpθT 1 + supσisin[0T]

uϵ(σ)

p

H1113888 1113889

(65)

So due to (30) we get

E suptisin[0T]

Ψ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (66)

Hence if we choose θ ≔ 18and1113957θand1113954θ thanks to (60)ndash(62)and (66) for any pge 2 and θlt θ we get

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (67)

Using the Holder inequality we can estimate (58) forp 1

Lemma 3 Under (A1)ndash(A4) for any θ isin [0 θ) and 0le hle 1there exists β(θ)gt 0 such that for anyTgt 0 pge 1 x isin D((minus A1)

θ) y isin H and t isin [0 T] it holds

supϵisin(01]

E uϵ(t) minus uϵ(t + h)

p

H

le cpθT hβ(θ)p

+ h1113872 1113873 1 +xp

θ +yp

H1113872 1113873

(68)

Proof For any tge 0 with t t + h isin [0 T] we have

uϵ(t + h) minus uϵ(t) U1(t + h t) minus I( 1113857uϵ(t) + ψ1 uϵ t( 1113857(t + h) + 1113946t+h

tU1(t + h r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t+h

tU1(t + h r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t+h

t1113946Z

U1(t + h r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz) ≔ 11139445

i1I

it

(69)

By proceeding as the proof of Proposition 44 in [12] and(60) fix θ isin [0 θ) for any pge 1 and it is possible to showthat

E I1t

p

Hle cpθTh

θp 1 +xp

θ +yp

H1113872 1113873

E I2t

p

Hle cpTh

pminus 1 1 +xp

θ +yp

H1113872 1113873

E I3t

p

Hle cpTh

p 1 +xp

θ +yp

H1113872 1113873

E I4t

p

Hle cpTh

(pminus 2)2minus β1 ρ1minus 2( )ρ1( )(p2) 1 +xp

θ +yp

H1113872 1113873

(70)

According to the proof of (39) using the Holder in-equality and (30) we have

E I5t

p

Hle cpE 1113946

t+h

t1 + uϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+ cpE1113946t+h

t1 + uϵ(r)

p

H1113872 1113873dr le cp h

(pminus 2)2+ 11113872 1113873

middot 1113946t+h

t1 + E uϵ(r)

p

H1113872 1113873dr

le cpT hp

2 + h1113874 1113875 1 +xp

H +yp

H1113872 1113873

(71)

en if we take pgt 1 such that

β1 ρ1 minus 2( 1113857

ρ1

p

p minus 2lt 1 (72)

We can get

Complexity 9

E uϵ(t + h) minus uϵ(t)

p

H

le cpT h(1minus 1p )p

+ hp

+ h(12)minus (1p)minus β1 ρ1minus 2( )2ρ1( )p

+ hp2

+ h1113874 1113875

times 1 +xp

H +yp

H1113872 1113873 + cpθThθp 1 +x

p

θ +yp

H1113872 1113873

(73)

As we are assuming |h|le 1 (68) follows for any pgep bytaking

β(θ) ≔ min θ 1 minus1p

112

minus1p

minusβ1 ρ1 minus 2( 1113857

2ρ112

1113896 1113897 (74)

From the Holder inequality we can estimate (68) forpltp

E uϵ(t + h) minus uϵ(t)

p

Hle E uϵ(t + h) minus uϵ(t)

p

H1113876 1113877

pp

(75)

So we have (68)According to the above lemma we get that for everyϵ isin (0 1] the function uϵ(t) is uniformly bounded aboutt isin [0 T] and it is also equicontinuous at every point oft isin [0 T] In view of eorem 123 in [43] we can infer thatthe set uϵ1113864 1113865ϵisin(01] is relatively compact in D([0 T]H) Inaddition according to eorem 132 in [43] and the abovelemma by using Chebyshevrsquos inequality we also can get thatthe family of probability measures L(uϵ)1113864 1113865ϵisin(01] is tight inP(D([0 T]H))

4 An Evolution Family of Measures for theFast Equation

For any frozen slow component x isin H any initial conditiony isin H and any s isin R we introduce the following problem

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) v(s) y

(76)

where

wQ2(t)

wQ21 (t)

wQ22 (minus t)

⎧⎨

⎩

if tge 0

if tlt 0

1113957N2prime(t z)

1113957N1prime(t z)

1113957N3prime(minus t z)

⎧⎪⎪⎨

⎪⎪⎩

if tge 0

if tlt 0

(77)

for two independent Q2-Wiener processes wQ21 (t) and

wQ22 (t) and two independent compensated Poisson mea-

sures 1113957N1prime(dt dz) and 1113957N3prime(dt dz) with the same Levymeasure are both defined as in Section 2

According to the definition of the operator ψα2(middot s) weknow that the mapping ψα2(middot s) C([s T] H)⟶C([s T]H) is a linearly bounded operator and it is Lip-schitz continuous Hence we have that for anyx y isin H pge 1 and sltT there exists a unique mild solution

vx(middot s y) isin Lp(Ω D([0 T]H)) (see [28]) in the followingform

vx(t s y) Uα2(t s)y + ψα2 v

x(middot s y) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s y)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s y)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s y) z( 1113857 1113957N2prime(dr dz)

(78)

Moreover if the space D(RH) endowed with the to-pology of uniform convergence on bounded intervals anFt1113864 1113865tisinR-adapted process vx isin Lp(Ω D([0 T]H)) is a mildsolution of the equation

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt

+ F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz)

(79)

where t isin R en for every slt t we have

vx(t) Uα2(t s)v

x(s) + ψα2 v

x s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r) z( 1113857 1113957N2prime

(dr dz)

(80)

In the following for any x isin H and any adapted processv we set

Γα(v s)(t) ≔ 1113946t

sUα2(t r)F2(r x v(r))dw

Q2(r) tgt s

Ψα(v s)(t) ≔ 1113946t

s1113946Z

Uα2(t r)G2(r x v(r) z) 1113957N2prime(dr dz) tgt s

(81)

For any 0lt δ lt α and any v1 and v2 with slt t by pro-ceeding as in the proof of Lemma 71 in [44] it is possible toshow that there exists pgt 1 such that for any pgep we have

suprisin[st]

eδp(rminus s)

E Γα v1 s( 1113857(r) minus Γα v2 s( 1113857(r)

p

H

le cp1L

p

f2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(82)

where Lf2is the Lipschitz constant of f2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 and slt tFor the stochastic term Ψα(v s)(t) using Kunitarsquos first

inequality ([26] eorem 4423) we get

10 Complexity

E Ψα v1 s( 1113857(t) minus Ψα v2 s( 1113857(t)

p

Hle cpE 1113946

t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

p

Hv2prime(dz)dr

le cpE 1113946t

s1113946Z

eminus 2α(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

2H

v2prime(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus αp(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

p

Hv2prime(dz)dr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)e

minus 2δ(tminus s)e2δ(rminus s)

E v1(r) minus v2(r)

2Hdr1113888 1113889

p2

+ cpLpg2

1113946t

se

minus p(αminus δ)(tminus r)e

minus δp(tminus s)eδp(rminus s)

E v1(r) minus v2(r)

p

Hdr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)dr1113888 1113889

p2

+ 1113946t

se

minus p(αminus δ)(tminus r)dr⎡⎣ ⎤⎦ times eminus δp(tminus s) sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

le cp1L

pg2

(α minus δ)cp2e

minus δp(tminus s) suprisin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(83)

so

suprisin[st]

eδp(rminus s)

E Ψα v1 s( 1113857(r) minus Ψα v2 s( 1113857(r)

p

Hle cp1

Lpg2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H (84)

where Lg2is the Lipschitz constant of g2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 andslt t

Moreover using (A4) we can show that

suprisin[st]

eδp(rminus s)

E Γα(v s)(r)

p

Hle cp1

Mp

f2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(85)

suprisin[st]

eδp(rminus s)

E Ψα(v s)(r)

p

Hle cp1

Mpg2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(86)

where Mf2andMg2

are the linear growth constants off2 andg2 and cp1 and cp2 are two suitable positive constantsindependent of αgt 0 and slt t

For any fixed adapted process v let us introduce theproblem

dρ(t) A2(t) minus α( 1113857ρ(t)dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) ρ(s) 0

(87)

We denote that its unique mild solution is ρα(v s) ismeans that ρα(v s) solves the equation

ρα(v s)(t) ψα2 ρα(v s) s( 1113857(t) + Γα(v s)(t)

+ Ψα(v s)(t) slt tltT(88)

Due to (82) and (84) using the same argument as (58) in[25] it is easy to prove that for any process v1 v2 and0lt δ lt α we have

suprisin[st]

eδp(rminus s)

E ρα v1 s( 1113857(r) minus ρα v2 s( 1113857(r)

p

H

le cp1Lp

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(89)

where L max Lb2 Lf2

Lg21113966 1113967

Similarly thanks to (85) and (86) for any process v and0lt δ lt α we can prove that

suprisin[st]

eδp(rminus s)

E ρα(v s)(r)

p

Hle cp1

Mp

(α minus δ)cp2

suprisin[st]

eδp(rminus s)

E 1 +v(r)p

H1113872 1113873

(90)

where M max Mb2 Mf2

Mg21113966 1113967

Complexity 11

Lemma 4 Under (A1)ndash(A4) there exists δ gt 0 such that forany x y isin H and pge 1

E vx(t s y)

p

Hle cp 1 +x

p

H + eminus δp(tminus s)

yp

H1113872 1113873 slt t

(91)

Proof We set Λα(t) ≔ vx(t s y) minus ρα(t) whereρα(t) ρα(vx(middot s y) s)(t) is the solution of problem (87)with v vx(middot s y) Using Youngrsquos inequality we have1p

ddtΛα(t)

p

Hlelang A2(t) minus α( 1113857Λα(t)Λα(t)rangH Λα(t)

pminus 2H

+langB2 t xΛα(t) + ρα(t)( 1113857 minus B2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2H

+langB2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2Hle minus α Λα(t)

p

H

+ c Λα(t)

p

H+ c 1 +xH + ρα(t)

H1113872 1113873 Λα(t)

pminus 1H

le minus α Λα(t)

p

H+ cp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(92)

Because α is large enough we can find η α minus cp gt 0such that

ddtΛα(t)

p

Hle minus ηp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(93)

According to the Gronwall inequality we have

Λα(t)

p

Hle e

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(94)

So for any pge 1

vx(t s y)

p

Hle cp ρα(t)

p

H+ cpe

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(95)

Fixing 0lt δ lt η according to (90) we get

eδp(tminus s)

E vx(t s y)

p

Hle cpe

δp(tminus s)E ρα(t)

p

H+ cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873 + cp 1113946t

seδp(rminus s)

E ρα(r)

p

Hdr

le cp1Mp

(α minus δ)cp2sup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873 + cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ cp1Mp

(α minus δ)cp21113946

t

ssup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873dr

(96)

Taking α1 (2cp1Mp)1cp2 + δ when αge α1 we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ 1113946t

ssup

risin[st]

eδp(rminus s)

E vx(r s y)

p

Hdr

(97)

Due to the Gronwall lemma we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

(98)

Hence we get (91)

Lemma 5 Under (A1)ndash(A4) for any t isin R and x y isin H forall pge 1 there exists ηx(t) isin Lp(ΩH) such that

lims⟶minus infin

E vx(t s y) minus ηx

(t)

p

H 0 (99)

Moreover for any pge 1 there exists some δp gt 0 suchthat

E vx(t s y) minus ηx

(t)

p

Hle cpe

minus δp(tminus s) 1 +xp

H +yp

H1113872 1113873

(100)

Finally ηx is a mild solution in R of (79)

Proof Fix hgt 0 and define

ρ(t) vx(t s y) minus v

x(t s minus h y) slt t (101)

We know that ρ(t) is the unique mild solution of theproblem

12 Complexity

dρ(t) A2(t) minus α( 1113857ρ(t) + B2 t x vx(t s y)( 1113857 minus B2 t x vx(t s minus h y)( 11138571113858 1113859dt

+ F2 t x vx(t s y)( 1113857 minus F2 t x vx(t s minus h y)( 11138571113858 1113859dwQ2(t)

+1113946Z

G2 t x vx(t s y)( 1113857 minus G2 t x vx(t s minus h y)( 11138571113858 1113859 1113957N2prime(dt dz)

ρ(s) y minus vx(s s minus h y)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(t) Uα2(t s) y minus vx(s s minus h y)( 1113857 + ψα2(ρ s)(t)

+ 1113946t

sUα2(t r) B2 r x v

x(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

+ 1113946t

sUα2(t r) F2 r x v

x(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r) G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime

(dr dz)

(102)

Multiply both sides of the above equation by eδp(tminus s)Because α large enough according to Lemma 24 in [25] wehave

eδp(tminus s)

Eρ(t)p

H le cpe(δminus α)p(tminus s)

E ec2(ts)A2 y minus v

x(s s minus h y)( 1113857

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

B2 r x vx(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

p

H

+ cpE 1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime(dr dz)

p

H

≔ 11139444

i1I

it

(103)

According to Lemma 31 in [12] we know that for anyJ isinL(Linfin(D)H)capL(H L1(D)) with J Jlowast and sge 0 wehave

esAi JQi

22 leKis

minus βi ρi minus 2( )ρi eminus α ρi+2( )ρi( )s

J2L Linfin(D)H( )

(104)

where

Ki βi

e1113888 1113889

βi ρi minus 2( )ρi

ζ ρi minus 2( )ρi

i κ2ρi

i (105)

Taking pgt 1 such that pβ2(ρ2 minus 2)[ρ2(p minus 2)]lt 1en using the BurkholderndashDavisndashGundy inequality andKunitarsquos first inequality we can get that for any pgep and0lt δ lt α it yields

I2t le cpL

p

b2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se

(δminus α)(tminus r)dr1113888 1113889

p

le cp

Lp

b2

(α minus δ)p suprisin[st]

eδp(rminus s)

Eρ(r)p

H (106)

Complexity 13

I3t le cpE 1113946

t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859Q2

2

2dr1113888 1113889

p2

le cpLp

f2K

p22 1113946

t

se2(δminus α)(tminus r)

c2(t r)minus β2 ρ2minus 2( )ρ2( )e

minus α ρ2+2( )ρ2c2(tr)dr1113888 1113889

p2

times suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se2(δminus α)(tminus r)

c0(t minus r)1113858 1113859minus β2 ρ2minus 2( )ρ2( )dr1113888 1113889

p2

le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0r

minus β2 ρ2minus 2( )ρ2( )(p(pminus 2))dr1113888 1113889

(pminus 2)2

1113946tminus s

0e

minus p(αminus δ)rdr1113888 1113889

le cp

Lp

f2

α minus δsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

I4t le cpE 1113946

t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

p

Hv2prime

(dz)dr

le cpLpg2

suprisin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0e

minus 2(αminus δ)rdr1113888 1113889

p2

+ 1113946tminus s

0e

minus p(αminus δ)rdr⎡⎣ ⎤⎦le cp

Lpg2

(α minus δ)cpsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(107)

Hence we have

suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H

+ cp1Lp

(α minus δ)cp2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(108)

erefore for αgt 0 large enough we can find 0lt δp lt αsuch that

cp1L

α minus δp1113872 1113873cp2lt 1 (109)

is implies that

suprisin[st]

epδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H (110)

Let δp pδp thanks to Lemma 4 we have

E vx(t s y) minus v

x(t s minus h y)

p

H

le cpeminus δp(tminus s)

y minus vx(s s minus h y)

p

H

le cpeminus δp(tminus s) 1 +x

p

H +yp

H + eminus δph

yp

H1113872 1113873

(111)

Because Lp(ΩH) has completeness for any pgep lets⟶ minus infin in (111) there exists ηx(t) isin Lp(ΩH) such that(99) holds en if we let h⟶infin in (111) we obtain (100)Using the Holder inequality we can get (99) and (100) holdsfor any pltp

If we take y1 y2 isin H use the same arguments forvx(t s y1) minus vx(t s y2) slt t we have

E vx

t s y1( 1113857 minus vx

t s y2( 1113857

p

Hle cpe

minus δp(tminus s)y1 minus y2

p

H slt t

(112)

Let s⟶ minus infin this means that the limit ηx(t) does notdepend on the initial condition y isin H

Finally we prove that ηx(t) is a mild solution of (79)Due to the limit ηx(t) does not depend on the initialcondition we can let initial condition y 0 For any slt t

and hgt 0 we have

vx(t s minus h 0) Uα2(t s)v

x(s s minus h 0) + ψα2 v

x(middot s minus h 0) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s minus h 0)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s minus h 0)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s minus h 0) z( 1113857 1113957N2prime

(dr dz)

(113)

Let h⟶infin on both sides due to (99) we can get forany slt t have

ηx(t) Uα2(t s)ηx

(s) + ψα2 ηx s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x ηx

(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x ηx

(r)( 1113857dwQ2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x ηx(r) z( 1113857 1113957N2prime

(dr dz)

(114)

is means that ηx(t) is a mild solution of (79)For any t isin R and x isin H we denote that the law of the

random variable ηx(t) is μxt and we introduce the transition

evolution operator

Pxstφ(y) Eφ v

x(t s y)( 1113857 slt t y isin H (115)

14 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

According to the definition of Λ2ϵ(t) for any t isin [0 T]we have

E vϵ(t)

p

Hle cpE Γ2ϵ(t)

p

H+ cpE Ψ2ϵ(t)

p

H+ cpe

minus (αp2ϵ)ty

p

H

+cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r) 1 + E uϵ(r)

p

H+ E Γ2ϵ(r)

p

H1113872

+ E Ψ2ϵ(r)

p

H1113873dr

(47)

erefore by integrating with respect to t using Youngrsquosinequality we obtain

1113946t

0E vϵ(r)

p

Hdrle cp(t)y

p

H + cp 1113946t

0E Γ2ϵ(r)

p

Hdr + 1113946

t

0E Ψ2ϵ(r)

p

Hdr1113888 1113889 +

cp

ϵ1113946

t

01 + E uϵ(r)

p

H+ E Γ2ϵ(r)

p

H+ E Ψ2ϵ(r)

p

H1113872 1113873dr

times 1113946t

0e

minus (αp2ϵ)rdrle cp(t) 1 +yp

H1113872 1113873 + cp 1113946t

0E uϵ(r)

p

Hdr + 1113946

t

0E Γ2ϵ(r)

p

Hdr + 1113946

t

0E Ψ2ϵ(r)

p

Hdr1113888 1113889

(48)

According to the BurkholderndashDavisndashGundy inequalityby proceeding as Proposition 42 in [12] we can easily get

1113946t

0E Γ2ϵ(r)

p

Hdrle cp(t) 1113946

t

01 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(49)

Concerning the stochastic term Ψ2ϵ(t) using Kunitarsquosfirst inequality we have

E Ψ2ϵ(t)

p

Hle cpE

1ϵ

1113946t

01113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2G2 r uϵ(r) vϵ(r) z( 1113857

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

01113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2G2 r uϵ(r) vϵ(r) z( 1113857

p

Hv2(dz)dr

lecp

ϵp2E1113946

t

0e

minus (α(pminus 2)ϵ)(tminus r) 1 + uϵ(r)

p

H+ vϵ(r)

p

H1113872 1113873dr 1113946

t

0e

minus (αpϵ)(pminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵE1113946

t

0e

minus (αpϵ)(tminus r) 1 + uϵ(r)

p

H+ vϵ(r)

p

H1113872 1113873drle

cp

ϵ1113946

t

0e

minus (αp2ϵ)(tminus r) 1 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(50)

By integrating with respect to t both sides and usingYoungrsquos inequality we have

1113946t

0E Ψ2ϵ(r)

p

Hdrle

cp

ϵ1113946

t

0e

minus (αp2ϵ)rdr middot 1113946t

01 + E uϵ(r)

p

H1113872

+ E vϵ(r)

p

H1113873drle cp(t) 1113946

t

01 + E uϵ(r)

p

H+ E vϵ(r)

p

H1113872 1113873dr

(51)

Substituting (49) and (51) into (48) we get

1113946t

0E vϵ(r)

p

Hdrle cp(t) 1 +y

p

H + 1113946t

0E uϵ(r)

p

Hdr1113888 1113889

+ cpT(t) 1113946t

0E vϵ(r)

p

Hdr

(52)

As cp(0) 0 and cp(t) is a continuous increasingfunction we can fix t0 gt 0 such that for any tle t0 we havecp(t)le 12 so

Complexity 7

1113946t

0E vϵ(r)

p

Hdrle cp(t) 1 +y

p

H + E suprisin[0t]

uϵ(r)

p

H1113888 1113889 t isin 0 t01113858 1113859

(53)

Due to (41) and (53) for any t isin [0 t0] we can get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT(t)E suprisin[0t]

uϵ(r)

p

H+ cpT 1113946

t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr

(54)

Similarly we also can fix 0lt t1 le t0 such that for anytle t1 we have cpT(t)le 12 so

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT 1113946t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr t isin 0 t11113858 1113859

(55)

According to the Gronwall inequality we get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(56)

Substituting (56) into (53) it yields

1113946t

0E vϵ(r)

p

Hdrle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(57)

For any pge 2 by repeating this in the intervals[t1 2t1] [2t1 3t1] etc we can easily get (31) Substituting(31) into (41) and using the Gronwall inequality again we get(30) Using the Holder inequality we can estimate (30) and(31) for p 1

Lemma 2 Under (A1)ndash(A4) there exists θgt 0 such that forany Tgt 0 pge 1 x isin D((minus A1)

θ) with θ isin [0 θ) and y isin Hthere exist a positive constant cpθT gt 0 such that

supϵisin(01]

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (58)

Proof Assuming that x isin D((minus A1)θ)(θ ge 0) for any

t isin [0 T] we have

uϵ(t) U1(t 0)x + ψ1 uϵ 0( 1113857(t)

+ 1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

(59)

Concerning the second term ψ1(uϵ 0)(t) we get

ψ1 uϵ 0( 1113857(t)

p

θ le cp 1113946t

0minus A1( 1113857

θe

c1(tr)A1L1(r)uϵ(r)dr

p

H

le cpθ 1113946t

0(t minus r)

minus θL1(r)uϵ(r)

Hdr1113888 1113889

p

le cpθ suprisin[0T]

uϵ(r)

p

H1113946

t

0(t minus r)

minus θdr1113888 1113889

p

le cpθT 1 +xp

H +yp

H1113872 1113873

(60)

For any pge 2 according to the proof of Proposition 43in [12] and thanks to (30) and (31) it is possible to show thatthere exists a 1113957θ ge 0 such that for any θle 1113957θand12 we have

E suptisin[0T]

1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

p

θle cpθT 1 +x

p

H +yp

H1113872 1113873

(61)

E suptisin[0T]

Γ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (62)

Concerning the stochastic term Ψ1ϵ(t) using the fac-torization argument we have

Ψ1ϵ(t) cθ 1113946t

0(t minus r)

θminus 1e

c1(tr)A1ϕϵθ(r)dr (63)

where

ϕϵθ(r) 1113946r

01113946Z

(r minus σ)minus θ

ec1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

(64)

Next for any pge 2 let 1113954θ ((14) minus (12p))and12p forany θle 1113954θ according to (A3) and Lemma 1 using Kunitarsquosfirst inequality and the Holder inequality we get

8 Complexity

Ψ1ϵ(t)

p

θ le cθ 1113946t

0(t minus r)

θminus 1 ϕϵθ(r)

θdr1113888 1113889

p

le cθ suprisin[0t]

ϕϵθ(r)

p

θ 1113946t

0(t minus r)

θminus 1dr1113888 1113889

p

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus θ

minus A1( 1113857θe

c1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

p

H

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus 2θ

(r minus σ)minus 2θ

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)dσ1113874 1113875p2

+ cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus pθ

(r minus σ)minus pθ

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)dσ le cpθT sup

risin[0t]

1113946r

01113946Z

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)1113874 1113875p2

dσ1113890

times 1113946r

0(r minus σ)

minus 4pθ(pminus 2)dσ1113874 1113875(pminus 2)2

1113891 + cpθT suprisin[0t]

1113946r

0(r minus σ)

minus 2pθdσ middot supσisin[0r]

1113946Z

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)1113888 11138891113890 1113891

le cpθT 1 + supσisin[0T]

uϵ(σ)

p

H1113888 1113889

(65)

So due to (30) we get

E suptisin[0T]

Ψ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (66)

Hence if we choose θ ≔ 18and1113957θand1113954θ thanks to (60)ndash(62)and (66) for any pge 2 and θlt θ we get

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (67)

Using the Holder inequality we can estimate (58) forp 1

Lemma 3 Under (A1)ndash(A4) for any θ isin [0 θ) and 0le hle 1there exists β(θ)gt 0 such that for anyTgt 0 pge 1 x isin D((minus A1)

θ) y isin H and t isin [0 T] it holds

supϵisin(01]

E uϵ(t) minus uϵ(t + h)

p

H

le cpθT hβ(θ)p

+ h1113872 1113873 1 +xp

θ +yp

H1113872 1113873

(68)

Proof For any tge 0 with t t + h isin [0 T] we have

uϵ(t + h) minus uϵ(t) U1(t + h t) minus I( 1113857uϵ(t) + ψ1 uϵ t( 1113857(t + h) + 1113946t+h

tU1(t + h r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t+h

tU1(t + h r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t+h

t1113946Z

U1(t + h r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz) ≔ 11139445

i1I

it

(69)

By proceeding as the proof of Proposition 44 in [12] and(60) fix θ isin [0 θ) for any pge 1 and it is possible to showthat

E I1t

p

Hle cpθTh

θp 1 +xp

θ +yp

H1113872 1113873

E I2t

p

Hle cpTh

pminus 1 1 +xp

θ +yp

H1113872 1113873

E I3t

p

Hle cpTh

p 1 +xp

θ +yp

H1113872 1113873

E I4t

p

Hle cpTh

(pminus 2)2minus β1 ρ1minus 2( )ρ1( )(p2) 1 +xp

θ +yp

H1113872 1113873

(70)

According to the proof of (39) using the Holder in-equality and (30) we have

E I5t

p

Hle cpE 1113946

t+h

t1 + uϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+ cpE1113946t+h

t1 + uϵ(r)

p

H1113872 1113873dr le cp h

(pminus 2)2+ 11113872 1113873

middot 1113946t+h

t1 + E uϵ(r)

p

H1113872 1113873dr

le cpT hp

2 + h1113874 1113875 1 +xp

H +yp

H1113872 1113873

(71)

en if we take pgt 1 such that

β1 ρ1 minus 2( 1113857

ρ1

p

p minus 2lt 1 (72)

We can get

Complexity 9

E uϵ(t + h) minus uϵ(t)

p

H

le cpT h(1minus 1p )p

+ hp

+ h(12)minus (1p)minus β1 ρ1minus 2( )2ρ1( )p

+ hp2

+ h1113874 1113875

times 1 +xp

H +yp

H1113872 1113873 + cpθThθp 1 +x

p

θ +yp

H1113872 1113873

(73)

As we are assuming |h|le 1 (68) follows for any pgep bytaking

β(θ) ≔ min θ 1 minus1p

112

minus1p

minusβ1 ρ1 minus 2( 1113857

2ρ112

1113896 1113897 (74)

From the Holder inequality we can estimate (68) forpltp

E uϵ(t + h) minus uϵ(t)

p

Hle E uϵ(t + h) minus uϵ(t)

p

H1113876 1113877

pp

(75)

So we have (68)According to the above lemma we get that for everyϵ isin (0 1] the function uϵ(t) is uniformly bounded aboutt isin [0 T] and it is also equicontinuous at every point oft isin [0 T] In view of eorem 123 in [43] we can infer thatthe set uϵ1113864 1113865ϵisin(01] is relatively compact in D([0 T]H) Inaddition according to eorem 132 in [43] and the abovelemma by using Chebyshevrsquos inequality we also can get thatthe family of probability measures L(uϵ)1113864 1113865ϵisin(01] is tight inP(D([0 T]H))

4 An Evolution Family of Measures for theFast Equation

For any frozen slow component x isin H any initial conditiony isin H and any s isin R we introduce the following problem

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) v(s) y

(76)

where

wQ2(t)

wQ21 (t)

wQ22 (minus t)

⎧⎨

⎩

if tge 0

if tlt 0

1113957N2prime(t z)

1113957N1prime(t z)

1113957N3prime(minus t z)

⎧⎪⎪⎨

⎪⎪⎩

if tge 0

if tlt 0

(77)

for two independent Q2-Wiener processes wQ21 (t) and

wQ22 (t) and two independent compensated Poisson mea-

sures 1113957N1prime(dt dz) and 1113957N3prime(dt dz) with the same Levymeasure are both defined as in Section 2

According to the definition of the operator ψα2(middot s) weknow that the mapping ψα2(middot s) C([s T] H)⟶C([s T]H) is a linearly bounded operator and it is Lip-schitz continuous Hence we have that for anyx y isin H pge 1 and sltT there exists a unique mild solution

vx(middot s y) isin Lp(Ω D([0 T]H)) (see [28]) in the followingform

vx(t s y) Uα2(t s)y + ψα2 v

x(middot s y) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s y)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s y)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s y) z( 1113857 1113957N2prime(dr dz)

(78)

Moreover if the space D(RH) endowed with the to-pology of uniform convergence on bounded intervals anFt1113864 1113865tisinR-adapted process vx isin Lp(Ω D([0 T]H)) is a mildsolution of the equation

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt

+ F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz)

(79)

where t isin R en for every slt t we have

vx(t) Uα2(t s)v

x(s) + ψα2 v

x s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r) z( 1113857 1113957N2prime

(dr dz)

(80)

In the following for any x isin H and any adapted processv we set

Γα(v s)(t) ≔ 1113946t

sUα2(t r)F2(r x v(r))dw

Q2(r) tgt s

Ψα(v s)(t) ≔ 1113946t

s1113946Z

Uα2(t r)G2(r x v(r) z) 1113957N2prime(dr dz) tgt s

(81)

For any 0lt δ lt α and any v1 and v2 with slt t by pro-ceeding as in the proof of Lemma 71 in [44] it is possible toshow that there exists pgt 1 such that for any pgep we have

suprisin[st]

eδp(rminus s)

E Γα v1 s( 1113857(r) minus Γα v2 s( 1113857(r)

p

H

le cp1L

p

f2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(82)

where Lf2is the Lipschitz constant of f2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 and slt tFor the stochastic term Ψα(v s)(t) using Kunitarsquos first

inequality ([26] eorem 4423) we get

10 Complexity

E Ψα v1 s( 1113857(t) minus Ψα v2 s( 1113857(t)

p

Hle cpE 1113946

t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

p

Hv2prime(dz)dr

le cpE 1113946t

s1113946Z

eminus 2α(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

2H

v2prime(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus αp(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

p

Hv2prime(dz)dr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)e

minus 2δ(tminus s)e2δ(rminus s)

E v1(r) minus v2(r)

2Hdr1113888 1113889

p2

+ cpLpg2

1113946t

se

minus p(αminus δ)(tminus r)e

minus δp(tminus s)eδp(rminus s)

E v1(r) minus v2(r)

p

Hdr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)dr1113888 1113889

p2

+ 1113946t

se

minus p(αminus δ)(tminus r)dr⎡⎣ ⎤⎦ times eminus δp(tminus s) sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

le cp1L

pg2

(α minus δ)cp2e

minus δp(tminus s) suprisin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(83)

so

suprisin[st]

eδp(rminus s)

E Ψα v1 s( 1113857(r) minus Ψα v2 s( 1113857(r)

p

Hle cp1

Lpg2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H (84)

where Lg2is the Lipschitz constant of g2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 andslt t

Moreover using (A4) we can show that

suprisin[st]

eδp(rminus s)

E Γα(v s)(r)

p

Hle cp1

Mp

f2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(85)

suprisin[st]

eδp(rminus s)

E Ψα(v s)(r)

p

Hle cp1

Mpg2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(86)

where Mf2andMg2

are the linear growth constants off2 andg2 and cp1 and cp2 are two suitable positive constantsindependent of αgt 0 and slt t

For any fixed adapted process v let us introduce theproblem

dρ(t) A2(t) minus α( 1113857ρ(t)dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) ρ(s) 0

(87)

We denote that its unique mild solution is ρα(v s) ismeans that ρα(v s) solves the equation

ρα(v s)(t) ψα2 ρα(v s) s( 1113857(t) + Γα(v s)(t)

+ Ψα(v s)(t) slt tltT(88)

Due to (82) and (84) using the same argument as (58) in[25] it is easy to prove that for any process v1 v2 and0lt δ lt α we have

suprisin[st]

eδp(rminus s)

E ρα v1 s( 1113857(r) minus ρα v2 s( 1113857(r)

p

H

le cp1Lp

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(89)

where L max Lb2 Lf2

Lg21113966 1113967

Similarly thanks to (85) and (86) for any process v and0lt δ lt α we can prove that

suprisin[st]

eδp(rminus s)

E ρα(v s)(r)

p

Hle cp1

Mp

(α minus δ)cp2

suprisin[st]

eδp(rminus s)

E 1 +v(r)p

H1113872 1113873

(90)

where M max Mb2 Mf2

Mg21113966 1113967

Complexity 11

Lemma 4 Under (A1)ndash(A4) there exists δ gt 0 such that forany x y isin H and pge 1

E vx(t s y)

p

Hle cp 1 +x

p

H + eminus δp(tminus s)

yp

H1113872 1113873 slt t

(91)

Proof We set Λα(t) ≔ vx(t s y) minus ρα(t) whereρα(t) ρα(vx(middot s y) s)(t) is the solution of problem (87)with v vx(middot s y) Using Youngrsquos inequality we have1p

ddtΛα(t)

p

Hlelang A2(t) minus α( 1113857Λα(t)Λα(t)rangH Λα(t)

pminus 2H

+langB2 t xΛα(t) + ρα(t)( 1113857 minus B2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2H

+langB2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2Hle minus α Λα(t)

p

H

+ c Λα(t)

p

H+ c 1 +xH + ρα(t)

H1113872 1113873 Λα(t)

pminus 1H

le minus α Λα(t)

p

H+ cp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(92)

Because α is large enough we can find η α minus cp gt 0such that

ddtΛα(t)

p

Hle minus ηp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(93)

According to the Gronwall inequality we have

Λα(t)

p

Hle e

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(94)

So for any pge 1

vx(t s y)

p

Hle cp ρα(t)

p

H+ cpe

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(95)

Fixing 0lt δ lt η according to (90) we get

eδp(tminus s)

E vx(t s y)

p

Hle cpe

δp(tminus s)E ρα(t)

p

H+ cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873 + cp 1113946t

seδp(rminus s)

E ρα(r)

p

Hdr

le cp1Mp

(α minus δ)cp2sup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873 + cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ cp1Mp

(α minus δ)cp21113946

t

ssup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873dr

(96)

Taking α1 (2cp1Mp)1cp2 + δ when αge α1 we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ 1113946t

ssup

risin[st]

eδp(rminus s)

E vx(r s y)

p

Hdr

(97)

Due to the Gronwall lemma we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

(98)

Hence we get (91)

Lemma 5 Under (A1)ndash(A4) for any t isin R and x y isin H forall pge 1 there exists ηx(t) isin Lp(ΩH) such that

lims⟶minus infin

E vx(t s y) minus ηx

(t)

p

H 0 (99)

Moreover for any pge 1 there exists some δp gt 0 suchthat

E vx(t s y) minus ηx

(t)

p

Hle cpe

minus δp(tminus s) 1 +xp

H +yp

H1113872 1113873

(100)

Finally ηx is a mild solution in R of (79)

Proof Fix hgt 0 and define

ρ(t) vx(t s y) minus v

x(t s minus h y) slt t (101)

We know that ρ(t) is the unique mild solution of theproblem

12 Complexity

dρ(t) A2(t) minus α( 1113857ρ(t) + B2 t x vx(t s y)( 1113857 minus B2 t x vx(t s minus h y)( 11138571113858 1113859dt

+ F2 t x vx(t s y)( 1113857 minus F2 t x vx(t s minus h y)( 11138571113858 1113859dwQ2(t)

+1113946Z

G2 t x vx(t s y)( 1113857 minus G2 t x vx(t s minus h y)( 11138571113858 1113859 1113957N2prime(dt dz)

ρ(s) y minus vx(s s minus h y)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(t) Uα2(t s) y minus vx(s s minus h y)( 1113857 + ψα2(ρ s)(t)

+ 1113946t

sUα2(t r) B2 r x v

x(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

+ 1113946t

sUα2(t r) F2 r x v

x(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r) G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime

(dr dz)

(102)

Multiply both sides of the above equation by eδp(tminus s)Because α large enough according to Lemma 24 in [25] wehave

eδp(tminus s)

Eρ(t)p

H le cpe(δminus α)p(tminus s)

E ec2(ts)A2 y minus v

x(s s minus h y)( 1113857

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

B2 r x vx(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

p

H

+ cpE 1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime(dr dz)

p

H

≔ 11139444

i1I

it

(103)

According to Lemma 31 in [12] we know that for anyJ isinL(Linfin(D)H)capL(H L1(D)) with J Jlowast and sge 0 wehave

esAi JQi

22 leKis

minus βi ρi minus 2( )ρi eminus α ρi+2( )ρi( )s

J2L Linfin(D)H( )

(104)

where

Ki βi

e1113888 1113889

βi ρi minus 2( )ρi

ζ ρi minus 2( )ρi

i κ2ρi

i (105)

Taking pgt 1 such that pβ2(ρ2 minus 2)[ρ2(p minus 2)]lt 1en using the BurkholderndashDavisndashGundy inequality andKunitarsquos first inequality we can get that for any pgep and0lt δ lt α it yields

I2t le cpL

p

b2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se

(δminus α)(tminus r)dr1113888 1113889

p

le cp

Lp

b2

(α minus δ)p suprisin[st]

eδp(rminus s)

Eρ(r)p

H (106)

Complexity 13

I3t le cpE 1113946

t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859Q2

2

2dr1113888 1113889

p2

le cpLp

f2K

p22 1113946

t

se2(δminus α)(tminus r)

c2(t r)minus β2 ρ2minus 2( )ρ2( )e

minus α ρ2+2( )ρ2c2(tr)dr1113888 1113889

p2

times suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se2(δminus α)(tminus r)

c0(t minus r)1113858 1113859minus β2 ρ2minus 2( )ρ2( )dr1113888 1113889

p2

le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0r

minus β2 ρ2minus 2( )ρ2( )(p(pminus 2))dr1113888 1113889

(pminus 2)2

1113946tminus s

0e

minus p(αminus δ)rdr1113888 1113889

le cp

Lp

f2

α minus δsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

I4t le cpE 1113946

t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

p

Hv2prime

(dz)dr

le cpLpg2

suprisin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0e

minus 2(αminus δ)rdr1113888 1113889

p2

+ 1113946tminus s

0e

minus p(αminus δ)rdr⎡⎣ ⎤⎦le cp

Lpg2

(α minus δ)cpsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(107)

Hence we have

suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H

+ cp1Lp

(α minus δ)cp2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(108)

erefore for αgt 0 large enough we can find 0lt δp lt αsuch that

cp1L

α minus δp1113872 1113873cp2lt 1 (109)

is implies that

suprisin[st]

epδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H (110)

Let δp pδp thanks to Lemma 4 we have

E vx(t s y) minus v

x(t s minus h y)

p

H

le cpeminus δp(tminus s)

y minus vx(s s minus h y)

p

H

le cpeminus δp(tminus s) 1 +x

p

H +yp

H + eminus δph

yp

H1113872 1113873

(111)

Because Lp(ΩH) has completeness for any pgep lets⟶ minus infin in (111) there exists ηx(t) isin Lp(ΩH) such that(99) holds en if we let h⟶infin in (111) we obtain (100)Using the Holder inequality we can get (99) and (100) holdsfor any pltp

If we take y1 y2 isin H use the same arguments forvx(t s y1) minus vx(t s y2) slt t we have

E vx

t s y1( 1113857 minus vx

t s y2( 1113857

p

Hle cpe

minus δp(tminus s)y1 minus y2

p

H slt t

(112)

Let s⟶ minus infin this means that the limit ηx(t) does notdepend on the initial condition y isin H

Finally we prove that ηx(t) is a mild solution of (79)Due to the limit ηx(t) does not depend on the initialcondition we can let initial condition y 0 For any slt t

and hgt 0 we have

vx(t s minus h 0) Uα2(t s)v

x(s s minus h 0) + ψα2 v

x(middot s minus h 0) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s minus h 0)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s minus h 0)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s minus h 0) z( 1113857 1113957N2prime

(dr dz)

(113)

Let h⟶infin on both sides due to (99) we can get forany slt t have

ηx(t) Uα2(t s)ηx

(s) + ψα2 ηx s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x ηx

(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x ηx

(r)( 1113857dwQ2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x ηx(r) z( 1113857 1113957N2prime

(dr dz)

(114)

is means that ηx(t) is a mild solution of (79)For any t isin R and x isin H we denote that the law of the

random variable ηx(t) is μxt and we introduce the transition

evolution operator

Pxstφ(y) Eφ v

x(t s y)( 1113857 slt t y isin H (115)

14 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

1113946t

0E vϵ(r)

p

Hdrle cp(t) 1 +y

p

H + E suprisin[0t]

uϵ(r)

p

H1113888 1113889 t isin 0 t01113858 1113859

(53)

Due to (41) and (53) for any t isin [0 t0] we can get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT(t)E suprisin[0t]

uϵ(r)

p

H+ cpT 1113946

t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr

(54)

Similarly we also can fix 0lt t1 le t0 such that for anytle t1 we have cpT(t)le 12 so

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873

+ cpT 1113946t

0E sup

σisin[0r]

uϵ(σ)

p

Hdr t isin 0 t11113858 1113859

(55)

According to the Gronwall inequality we get

E suprisin[0t]

uϵ(r)

p

Hle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(56)

Substituting (56) into (53) it yields

1113946t

0E vϵ(r)

p

Hdrle cpT 1 +x

p

H +yp

H1113872 1113873 t isin 0 t11113858 1113859

(57)

For any pge 2 by repeating this in the intervals[t1 2t1] [2t1 3t1] etc we can easily get (31) Substituting(31) into (41) and using the Gronwall inequality again we get(30) Using the Holder inequality we can estimate (30) and(31) for p 1

Lemma 2 Under (A1)ndash(A4) there exists θgt 0 such that forany Tgt 0 pge 1 x isin D((minus A1)

θ) with θ isin [0 θ) and y isin Hthere exist a positive constant cpθT gt 0 such that

supϵisin(01]

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (58)

Proof Assuming that x isin D((minus A1)θ)(θ ge 0) for any

t isin [0 T] we have

uϵ(t) U1(t 0)x + ψ1 uϵ 0( 1113857(t)

+ 1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t

0U1(t r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t

01113946Z

U1(t r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz)

(59)

Concerning the second term ψ1(uϵ 0)(t) we get

ψ1 uϵ 0( 1113857(t)

p

θ le cp 1113946t

0minus A1( 1113857

θe

c1(tr)A1L1(r)uϵ(r)dr

p

H

le cpθ 1113946t

0(t minus r)

minus θL1(r)uϵ(r)

Hdr1113888 1113889

p

le cpθ suprisin[0T]

uϵ(r)

p

H1113946

t

0(t minus r)

minus θdr1113888 1113889

p

le cpθT 1 +xp

H +yp

H1113872 1113873

(60)

For any pge 2 according to the proof of Proposition 43in [12] and thanks to (30) and (31) it is possible to show thatthere exists a 1113957θ ge 0 such that for any θle 1113957θand12 we have

E suptisin[0T]

1113946t

0U1(t r)B1 r uϵ(r) vϵ(r)( 1113857dr

p

θle cpθT 1 +x

p

H +yp

H1113872 1113873

(61)

E suptisin[0T]

Γ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (62)

Concerning the stochastic term Ψ1ϵ(t) using the fac-torization argument we have

Ψ1ϵ(t) cθ 1113946t

0(t minus r)

θminus 1e

c1(tr)A1ϕϵθ(r)dr (63)

where

ϕϵθ(r) 1113946r

01113946Z

(r minus σ)minus θ

ec1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

(64)

Next for any pge 2 let 1113954θ ((14) minus (12p))and12p forany θle 1113954θ according to (A3) and Lemma 1 using Kunitarsquosfirst inequality and the Holder inequality we get

8 Complexity

Ψ1ϵ(t)

p

θ le cθ 1113946t

0(t minus r)

θminus 1 ϕϵθ(r)

θdr1113888 1113889

p

le cθ suprisin[0t]

ϕϵθ(r)

p

θ 1113946t

0(t minus r)

θminus 1dr1113888 1113889

p

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus θ

minus A1( 1113857θe

c1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

p

H

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus 2θ

(r minus σ)minus 2θ

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)dσ1113874 1113875p2

+ cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus pθ

(r minus σ)minus pθ

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)dσ le cpθT sup

risin[0t]

1113946r

01113946Z

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)1113874 1113875p2

dσ1113890

times 1113946r

0(r minus σ)

minus 4pθ(pminus 2)dσ1113874 1113875(pminus 2)2

1113891 + cpθT suprisin[0t]

1113946r

0(r minus σ)

minus 2pθdσ middot supσisin[0r]

1113946Z

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)1113888 11138891113890 1113891

le cpθT 1 + supσisin[0T]

uϵ(σ)

p

H1113888 1113889

(65)

So due to (30) we get

E suptisin[0T]

Ψ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (66)

Hence if we choose θ ≔ 18and1113957θand1113954θ thanks to (60)ndash(62)and (66) for any pge 2 and θlt θ we get

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (67)

Using the Holder inequality we can estimate (58) forp 1

Lemma 3 Under (A1)ndash(A4) for any θ isin [0 θ) and 0le hle 1there exists β(θ)gt 0 such that for anyTgt 0 pge 1 x isin D((minus A1)

θ) y isin H and t isin [0 T] it holds

supϵisin(01]

E uϵ(t) minus uϵ(t + h)

p

H

le cpθT hβ(θ)p

+ h1113872 1113873 1 +xp

θ +yp

H1113872 1113873

(68)

Proof For any tge 0 with t t + h isin [0 T] we have

uϵ(t + h) minus uϵ(t) U1(t + h t) minus I( 1113857uϵ(t) + ψ1 uϵ t( 1113857(t + h) + 1113946t+h

tU1(t + h r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t+h

tU1(t + h r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t+h

t1113946Z

U1(t + h r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz) ≔ 11139445

i1I

it

(69)

By proceeding as the proof of Proposition 44 in [12] and(60) fix θ isin [0 θ) for any pge 1 and it is possible to showthat

E I1t

p

Hle cpθTh

θp 1 +xp

θ +yp

H1113872 1113873

E I2t

p

Hle cpTh

pminus 1 1 +xp

θ +yp

H1113872 1113873

E I3t

p

Hle cpTh

p 1 +xp

θ +yp

H1113872 1113873

E I4t

p

Hle cpTh

(pminus 2)2minus β1 ρ1minus 2( )ρ1( )(p2) 1 +xp

θ +yp

H1113872 1113873

(70)

According to the proof of (39) using the Holder in-equality and (30) we have

E I5t

p

Hle cpE 1113946

t+h

t1 + uϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+ cpE1113946t+h

t1 + uϵ(r)

p

H1113872 1113873dr le cp h

(pminus 2)2+ 11113872 1113873

middot 1113946t+h

t1 + E uϵ(r)

p

H1113872 1113873dr

le cpT hp

2 + h1113874 1113875 1 +xp

H +yp

H1113872 1113873

(71)

en if we take pgt 1 such that

β1 ρ1 minus 2( 1113857

ρ1

p

p minus 2lt 1 (72)

We can get

Complexity 9

E uϵ(t + h) minus uϵ(t)

p

H

le cpT h(1minus 1p )p

+ hp

+ h(12)minus (1p)minus β1 ρ1minus 2( )2ρ1( )p

+ hp2

+ h1113874 1113875

times 1 +xp

H +yp

H1113872 1113873 + cpθThθp 1 +x

p

θ +yp

H1113872 1113873

(73)

As we are assuming |h|le 1 (68) follows for any pgep bytaking

β(θ) ≔ min θ 1 minus1p

112

minus1p

minusβ1 ρ1 minus 2( 1113857

2ρ112

1113896 1113897 (74)

From the Holder inequality we can estimate (68) forpltp

E uϵ(t + h) minus uϵ(t)

p

Hle E uϵ(t + h) minus uϵ(t)

p

H1113876 1113877

pp

(75)

So we have (68)According to the above lemma we get that for everyϵ isin (0 1] the function uϵ(t) is uniformly bounded aboutt isin [0 T] and it is also equicontinuous at every point oft isin [0 T] In view of eorem 123 in [43] we can infer thatthe set uϵ1113864 1113865ϵisin(01] is relatively compact in D([0 T]H) Inaddition according to eorem 132 in [43] and the abovelemma by using Chebyshevrsquos inequality we also can get thatthe family of probability measures L(uϵ)1113864 1113865ϵisin(01] is tight inP(D([0 T]H))

4 An Evolution Family of Measures for theFast Equation

For any frozen slow component x isin H any initial conditiony isin H and any s isin R we introduce the following problem

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) v(s) y

(76)

where

wQ2(t)

wQ21 (t)

wQ22 (minus t)

⎧⎨

⎩

if tge 0

if tlt 0

1113957N2prime(t z)

1113957N1prime(t z)

1113957N3prime(minus t z)

⎧⎪⎪⎨

⎪⎪⎩

if tge 0

if tlt 0

(77)

for two independent Q2-Wiener processes wQ21 (t) and

wQ22 (t) and two independent compensated Poisson mea-

sures 1113957N1prime(dt dz) and 1113957N3prime(dt dz) with the same Levymeasure are both defined as in Section 2

According to the definition of the operator ψα2(middot s) weknow that the mapping ψα2(middot s) C([s T] H)⟶C([s T]H) is a linearly bounded operator and it is Lip-schitz continuous Hence we have that for anyx y isin H pge 1 and sltT there exists a unique mild solution

vx(middot s y) isin Lp(Ω D([0 T]H)) (see [28]) in the followingform

vx(t s y) Uα2(t s)y + ψα2 v

x(middot s y) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s y)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s y)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s y) z( 1113857 1113957N2prime(dr dz)

(78)

Moreover if the space D(RH) endowed with the to-pology of uniform convergence on bounded intervals anFt1113864 1113865tisinR-adapted process vx isin Lp(Ω D([0 T]H)) is a mildsolution of the equation

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt

+ F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz)

(79)

where t isin R en for every slt t we have

vx(t) Uα2(t s)v

x(s) + ψα2 v

x s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r) z( 1113857 1113957N2prime

(dr dz)

(80)

In the following for any x isin H and any adapted processv we set

Γα(v s)(t) ≔ 1113946t

sUα2(t r)F2(r x v(r))dw

Q2(r) tgt s

Ψα(v s)(t) ≔ 1113946t

s1113946Z

Uα2(t r)G2(r x v(r) z) 1113957N2prime(dr dz) tgt s

(81)

For any 0lt δ lt α and any v1 and v2 with slt t by pro-ceeding as in the proof of Lemma 71 in [44] it is possible toshow that there exists pgt 1 such that for any pgep we have

suprisin[st]

eδp(rminus s)

E Γα v1 s( 1113857(r) minus Γα v2 s( 1113857(r)

p

H

le cp1L

p

f2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(82)

where Lf2is the Lipschitz constant of f2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 and slt tFor the stochastic term Ψα(v s)(t) using Kunitarsquos first

inequality ([26] eorem 4423) we get

10 Complexity

E Ψα v1 s( 1113857(t) minus Ψα v2 s( 1113857(t)

p

Hle cpE 1113946

t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

p

Hv2prime(dz)dr

le cpE 1113946t

s1113946Z

eminus 2α(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

2H

v2prime(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus αp(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

p

Hv2prime(dz)dr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)e

minus 2δ(tminus s)e2δ(rminus s)

E v1(r) minus v2(r)

2Hdr1113888 1113889

p2

+ cpLpg2

1113946t

se

minus p(αminus δ)(tminus r)e

minus δp(tminus s)eδp(rminus s)

E v1(r) minus v2(r)

p

Hdr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)dr1113888 1113889

p2

+ 1113946t

se

minus p(αminus δ)(tminus r)dr⎡⎣ ⎤⎦ times eminus δp(tminus s) sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

le cp1L

pg2

(α minus δ)cp2e

minus δp(tminus s) suprisin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(83)

so

suprisin[st]

eδp(rminus s)

E Ψα v1 s( 1113857(r) minus Ψα v2 s( 1113857(r)

p

Hle cp1

Lpg2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H (84)

where Lg2is the Lipschitz constant of g2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 andslt t

Moreover using (A4) we can show that

suprisin[st]

eδp(rminus s)

E Γα(v s)(r)

p

Hle cp1

Mp

f2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(85)

suprisin[st]

eδp(rminus s)

E Ψα(v s)(r)

p

Hle cp1

Mpg2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(86)

where Mf2andMg2

are the linear growth constants off2 andg2 and cp1 and cp2 are two suitable positive constantsindependent of αgt 0 and slt t

For any fixed adapted process v let us introduce theproblem

dρ(t) A2(t) minus α( 1113857ρ(t)dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) ρ(s) 0

(87)

We denote that its unique mild solution is ρα(v s) ismeans that ρα(v s) solves the equation

ρα(v s)(t) ψα2 ρα(v s) s( 1113857(t) + Γα(v s)(t)

+ Ψα(v s)(t) slt tltT(88)

Due to (82) and (84) using the same argument as (58) in[25] it is easy to prove that for any process v1 v2 and0lt δ lt α we have

suprisin[st]

eδp(rminus s)

E ρα v1 s( 1113857(r) minus ρα v2 s( 1113857(r)

p

H

le cp1Lp

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(89)

where L max Lb2 Lf2

Lg21113966 1113967

Similarly thanks to (85) and (86) for any process v and0lt δ lt α we can prove that

suprisin[st]

eδp(rminus s)

E ρα(v s)(r)

p

Hle cp1

Mp

(α minus δ)cp2

suprisin[st]

eδp(rminus s)

E 1 +v(r)p

H1113872 1113873

(90)

where M max Mb2 Mf2

Mg21113966 1113967

Complexity 11

Lemma 4 Under (A1)ndash(A4) there exists δ gt 0 such that forany x y isin H and pge 1

E vx(t s y)

p

Hle cp 1 +x

p

H + eminus δp(tminus s)

yp

H1113872 1113873 slt t

(91)

Proof We set Λα(t) ≔ vx(t s y) minus ρα(t) whereρα(t) ρα(vx(middot s y) s)(t) is the solution of problem (87)with v vx(middot s y) Using Youngrsquos inequality we have1p

ddtΛα(t)

p

Hlelang A2(t) minus α( 1113857Λα(t)Λα(t)rangH Λα(t)

pminus 2H

+langB2 t xΛα(t) + ρα(t)( 1113857 minus B2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2H

+langB2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2Hle minus α Λα(t)

p

H

+ c Λα(t)

p

H+ c 1 +xH + ρα(t)

H1113872 1113873 Λα(t)

pminus 1H

le minus α Λα(t)

p

H+ cp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(92)

Because α is large enough we can find η α minus cp gt 0such that

ddtΛα(t)

p

Hle minus ηp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(93)

According to the Gronwall inequality we have

Λα(t)

p

Hle e

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(94)

So for any pge 1

vx(t s y)

p

Hle cp ρα(t)

p

H+ cpe

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(95)

Fixing 0lt δ lt η according to (90) we get

eδp(tminus s)

E vx(t s y)

p

Hle cpe

δp(tminus s)E ρα(t)

p

H+ cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873 + cp 1113946t

seδp(rminus s)

E ρα(r)

p

Hdr

le cp1Mp

(α minus δ)cp2sup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873 + cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ cp1Mp

(α minus δ)cp21113946

t

ssup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873dr

(96)

Taking α1 (2cp1Mp)1cp2 + δ when αge α1 we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ 1113946t

ssup

risin[st]

eδp(rminus s)

E vx(r s y)

p

Hdr

(97)

Due to the Gronwall lemma we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

(98)

Hence we get (91)

Lemma 5 Under (A1)ndash(A4) for any t isin R and x y isin H forall pge 1 there exists ηx(t) isin Lp(ΩH) such that

lims⟶minus infin

E vx(t s y) minus ηx

(t)

p

H 0 (99)

Moreover for any pge 1 there exists some δp gt 0 suchthat

E vx(t s y) minus ηx

(t)

p

Hle cpe

minus δp(tminus s) 1 +xp

H +yp

H1113872 1113873

(100)

Finally ηx is a mild solution in R of (79)

Proof Fix hgt 0 and define

ρ(t) vx(t s y) minus v

x(t s minus h y) slt t (101)

We know that ρ(t) is the unique mild solution of theproblem

12 Complexity

dρ(t) A2(t) minus α( 1113857ρ(t) + B2 t x vx(t s y)( 1113857 minus B2 t x vx(t s minus h y)( 11138571113858 1113859dt

+ F2 t x vx(t s y)( 1113857 minus F2 t x vx(t s minus h y)( 11138571113858 1113859dwQ2(t)

+1113946Z

G2 t x vx(t s y)( 1113857 minus G2 t x vx(t s minus h y)( 11138571113858 1113859 1113957N2prime(dt dz)

ρ(s) y minus vx(s s minus h y)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(t) Uα2(t s) y minus vx(s s minus h y)( 1113857 + ψα2(ρ s)(t)

+ 1113946t

sUα2(t r) B2 r x v

x(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

+ 1113946t

sUα2(t r) F2 r x v

x(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r) G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime

(dr dz)

(102)

Multiply both sides of the above equation by eδp(tminus s)Because α large enough according to Lemma 24 in [25] wehave

eδp(tminus s)

Eρ(t)p

H le cpe(δminus α)p(tminus s)

E ec2(ts)A2 y minus v

x(s s minus h y)( 1113857

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

B2 r x vx(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

p

H

+ cpE 1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime(dr dz)

p

H

≔ 11139444

i1I

it

(103)

According to Lemma 31 in [12] we know that for anyJ isinL(Linfin(D)H)capL(H L1(D)) with J Jlowast and sge 0 wehave

esAi JQi

22 leKis

minus βi ρi minus 2( )ρi eminus α ρi+2( )ρi( )s

J2L Linfin(D)H( )

(104)

where

Ki βi

e1113888 1113889

βi ρi minus 2( )ρi

ζ ρi minus 2( )ρi

i κ2ρi

i (105)

Taking pgt 1 such that pβ2(ρ2 minus 2)[ρ2(p minus 2)]lt 1en using the BurkholderndashDavisndashGundy inequality andKunitarsquos first inequality we can get that for any pgep and0lt δ lt α it yields

I2t le cpL

p

b2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se

(δminus α)(tminus r)dr1113888 1113889

p

le cp

Lp

b2

(α minus δ)p suprisin[st]

eδp(rminus s)

Eρ(r)p

H (106)

Complexity 13

I3t le cpE 1113946

t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859Q2

2

2dr1113888 1113889

p2

le cpLp

f2K

p22 1113946

t

se2(δminus α)(tminus r)

c2(t r)minus β2 ρ2minus 2( )ρ2( )e

minus α ρ2+2( )ρ2c2(tr)dr1113888 1113889

p2

times suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se2(δminus α)(tminus r)

c0(t minus r)1113858 1113859minus β2 ρ2minus 2( )ρ2( )dr1113888 1113889

p2

le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0r

minus β2 ρ2minus 2( )ρ2( )(p(pminus 2))dr1113888 1113889

(pminus 2)2

1113946tminus s

0e

minus p(αminus δ)rdr1113888 1113889

le cp

Lp

f2

α minus δsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

I4t le cpE 1113946

t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

p

Hv2prime

(dz)dr

le cpLpg2

suprisin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0e

minus 2(αminus δ)rdr1113888 1113889

p2

+ 1113946tminus s

0e

minus p(αminus δ)rdr⎡⎣ ⎤⎦le cp

Lpg2

(α minus δ)cpsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(107)

Hence we have

suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H

+ cp1Lp

(α minus δ)cp2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(108)

erefore for αgt 0 large enough we can find 0lt δp lt αsuch that

cp1L

α minus δp1113872 1113873cp2lt 1 (109)

is implies that

suprisin[st]

epδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H (110)

Let δp pδp thanks to Lemma 4 we have

E vx(t s y) minus v

x(t s minus h y)

p

H

le cpeminus δp(tminus s)

y minus vx(s s minus h y)

p

H

le cpeminus δp(tminus s) 1 +x

p

H +yp

H + eminus δph

yp

H1113872 1113873

(111)

Because Lp(ΩH) has completeness for any pgep lets⟶ minus infin in (111) there exists ηx(t) isin Lp(ΩH) such that(99) holds en if we let h⟶infin in (111) we obtain (100)Using the Holder inequality we can get (99) and (100) holdsfor any pltp

If we take y1 y2 isin H use the same arguments forvx(t s y1) minus vx(t s y2) slt t we have

E vx

t s y1( 1113857 minus vx

t s y2( 1113857

p

Hle cpe

minus δp(tminus s)y1 minus y2

p

H slt t

(112)

Let s⟶ minus infin this means that the limit ηx(t) does notdepend on the initial condition y isin H

Finally we prove that ηx(t) is a mild solution of (79)Due to the limit ηx(t) does not depend on the initialcondition we can let initial condition y 0 For any slt t

and hgt 0 we have

vx(t s minus h 0) Uα2(t s)v

x(s s minus h 0) + ψα2 v

x(middot s minus h 0) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s minus h 0)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s minus h 0)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s minus h 0) z( 1113857 1113957N2prime

(dr dz)

(113)

Let h⟶infin on both sides due to (99) we can get forany slt t have

ηx(t) Uα2(t s)ηx

(s) + ψα2 ηx s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x ηx

(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x ηx

(r)( 1113857dwQ2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x ηx(r) z( 1113857 1113957N2prime

(dr dz)

(114)

is means that ηx(t) is a mild solution of (79)For any t isin R and x isin H we denote that the law of the

random variable ηx(t) is μxt and we introduce the transition

evolution operator

Pxstφ(y) Eφ v

x(t s y)( 1113857 slt t y isin H (115)

14 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

Ψ1ϵ(t)

p

θ le cθ 1113946t

0(t minus r)

θminus 1 ϕϵθ(r)

θdr1113888 1113889

p

le cθ suprisin[0t]

ϕϵθ(r)

p

θ 1113946t

0(t minus r)

θminus 1dr1113888 1113889

p

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus θ

minus A1( 1113857θe

c1(rσ)A1G1 σ uϵ(σ) z( 1113857 1113957N1(dσ dz)

p

H

le cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus 2θ

(r minus σ)minus 2θ

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)dσ1113874 1113875p2

+ cpθT suprisin[0t]

1113946r

01113946Z

(r minus σ)minus pθ

(r minus σ)minus pθ

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)dσ le cpθT sup

risin[0t]

1113946r

01113946Z

G1 σ uϵ(σ) z( 1113857

2H

v1(dz)1113874 1113875p2

dσ1113890

times 1113946r

0(r minus σ)

minus 4pθ(pminus 2)dσ1113874 1113875(pminus 2)2

1113891 + cpθT suprisin[0t]

1113946r

0(r minus σ)

minus 2pθdσ middot supσisin[0r]

1113946Z

G1 σ uϵ(σ) z( 1113857

p

Hv1(dz)1113888 11138891113890 1113891

le cpθT 1 + supσisin[0T]

uϵ(σ)

p

H1113888 1113889

(65)

So due to (30) we get

E suptisin[0T]

Ψ1ϵ(t)

p

θ le cpθT 1 +xp

H +yp

H1113872 1113873 (66)

Hence if we choose θ ≔ 18and1113957θand1113954θ thanks to (60)ndash(62)and (66) for any pge 2 and θlt θ we get

E suptisin[0T]

uϵ(t)

p

θ le cpθT 1 +xp

θ +yp

H1113872 1113873 (67)

Using the Holder inequality we can estimate (58) forp 1

Lemma 3 Under (A1)ndash(A4) for any θ isin [0 θ) and 0le hle 1there exists β(θ)gt 0 such that for anyTgt 0 pge 1 x isin D((minus A1)

θ) y isin H and t isin [0 T] it holds

supϵisin(01]

E uϵ(t) minus uϵ(t + h)

p

H

le cpθT hβ(θ)p

+ h1113872 1113873 1 +xp

θ +yp

H1113872 1113873

(68)

Proof For any tge 0 with t t + h isin [0 T] we have

uϵ(t + h) minus uϵ(t) U1(t + h t) minus I( 1113857uϵ(t) + ψ1 uϵ t( 1113857(t + h) + 1113946t+h

tU1(t + h r)B1 r uϵ(r) vϵ(r)( 1113857dr

+ 1113946t+h

tU1(t + h r)F1 r uϵ(r)( 1113857dw

Q1(r)

+ 1113946t+h

t1113946Z

U1(t + h r)G1 r uϵ(r) z( 1113857 1113957N1(dr dz) ≔ 11139445

i1I

it

(69)

By proceeding as the proof of Proposition 44 in [12] and(60) fix θ isin [0 θ) for any pge 1 and it is possible to showthat

E I1t

p

Hle cpθTh

θp 1 +xp

θ +yp

H1113872 1113873

E I2t

p

Hle cpTh

pminus 1 1 +xp

θ +yp

H1113872 1113873

E I3t

p

Hle cpTh

p 1 +xp

θ +yp

H1113872 1113873

E I4t

p

Hle cpTh

(pminus 2)2minus β1 ρ1minus 2( )ρ1( )(p2) 1 +xp

θ +yp

H1113872 1113873

(70)

According to the proof of (39) using the Holder in-equality and (30) we have

E I5t

p

Hle cpE 1113946

t+h

t1 + uϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+ cpE1113946t+h

t1 + uϵ(r)

p

H1113872 1113873dr le cp h

(pminus 2)2+ 11113872 1113873

middot 1113946t+h

t1 + E uϵ(r)

p

H1113872 1113873dr

le cpT hp

2 + h1113874 1113875 1 +xp

H +yp

H1113872 1113873

(71)

en if we take pgt 1 such that

β1 ρ1 minus 2( 1113857

ρ1

p

p minus 2lt 1 (72)

We can get

Complexity 9

E uϵ(t + h) minus uϵ(t)

p

H

le cpT h(1minus 1p )p

+ hp

+ h(12)minus (1p)minus β1 ρ1minus 2( )2ρ1( )p

+ hp2

+ h1113874 1113875

times 1 +xp

H +yp

H1113872 1113873 + cpθThθp 1 +x

p

θ +yp

H1113872 1113873

(73)

As we are assuming |h|le 1 (68) follows for any pgep bytaking

β(θ) ≔ min θ 1 minus1p

112

minus1p

minusβ1 ρ1 minus 2( 1113857

2ρ112

1113896 1113897 (74)

From the Holder inequality we can estimate (68) forpltp

E uϵ(t + h) minus uϵ(t)

p

Hle E uϵ(t + h) minus uϵ(t)

p

H1113876 1113877

pp

(75)

So we have (68)According to the above lemma we get that for everyϵ isin (0 1] the function uϵ(t) is uniformly bounded aboutt isin [0 T] and it is also equicontinuous at every point oft isin [0 T] In view of eorem 123 in [43] we can infer thatthe set uϵ1113864 1113865ϵisin(01] is relatively compact in D([0 T]H) Inaddition according to eorem 132 in [43] and the abovelemma by using Chebyshevrsquos inequality we also can get thatthe family of probability measures L(uϵ)1113864 1113865ϵisin(01] is tight inP(D([0 T]H))

4 An Evolution Family of Measures for theFast Equation

For any frozen slow component x isin H any initial conditiony isin H and any s isin R we introduce the following problem

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) v(s) y

(76)

where

wQ2(t)

wQ21 (t)

wQ22 (minus t)

⎧⎨

⎩

if tge 0

if tlt 0

1113957N2prime(t z)

1113957N1prime(t z)

1113957N3prime(minus t z)

⎧⎪⎪⎨

⎪⎪⎩

if tge 0

if tlt 0

(77)

for two independent Q2-Wiener processes wQ21 (t) and

wQ22 (t) and two independent compensated Poisson mea-

sures 1113957N1prime(dt dz) and 1113957N3prime(dt dz) with the same Levymeasure are both defined as in Section 2

According to the definition of the operator ψα2(middot s) weknow that the mapping ψα2(middot s) C([s T] H)⟶C([s T]H) is a linearly bounded operator and it is Lip-schitz continuous Hence we have that for anyx y isin H pge 1 and sltT there exists a unique mild solution

vx(middot s y) isin Lp(Ω D([0 T]H)) (see [28]) in the followingform

vx(t s y) Uα2(t s)y + ψα2 v

x(middot s y) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s y)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s y)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s y) z( 1113857 1113957N2prime(dr dz)

(78)

Moreover if the space D(RH) endowed with the to-pology of uniform convergence on bounded intervals anFt1113864 1113865tisinR-adapted process vx isin Lp(Ω D([0 T]H)) is a mildsolution of the equation

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt

+ F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz)

(79)

where t isin R en for every slt t we have

vx(t) Uα2(t s)v

x(s) + ψα2 v

x s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r) z( 1113857 1113957N2prime

(dr dz)

(80)

In the following for any x isin H and any adapted processv we set

Γα(v s)(t) ≔ 1113946t

sUα2(t r)F2(r x v(r))dw

Q2(r) tgt s

Ψα(v s)(t) ≔ 1113946t

s1113946Z

Uα2(t r)G2(r x v(r) z) 1113957N2prime(dr dz) tgt s

(81)

For any 0lt δ lt α and any v1 and v2 with slt t by pro-ceeding as in the proof of Lemma 71 in [44] it is possible toshow that there exists pgt 1 such that for any pgep we have

suprisin[st]

eδp(rminus s)

E Γα v1 s( 1113857(r) minus Γα v2 s( 1113857(r)

p

H

le cp1L

p

f2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(82)

where Lf2is the Lipschitz constant of f2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 and slt tFor the stochastic term Ψα(v s)(t) using Kunitarsquos first

inequality ([26] eorem 4423) we get

10 Complexity

E Ψα v1 s( 1113857(t) minus Ψα v2 s( 1113857(t)

p

Hle cpE 1113946

t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

p

Hv2prime(dz)dr

le cpE 1113946t

s1113946Z

eminus 2α(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

2H

v2prime(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus αp(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

p

Hv2prime(dz)dr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)e

minus 2δ(tminus s)e2δ(rminus s)

E v1(r) minus v2(r)

2Hdr1113888 1113889

p2

+ cpLpg2

1113946t

se

minus p(αminus δ)(tminus r)e

minus δp(tminus s)eδp(rminus s)

E v1(r) minus v2(r)

p

Hdr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)dr1113888 1113889

p2

+ 1113946t

se

minus p(αminus δ)(tminus r)dr⎡⎣ ⎤⎦ times eminus δp(tminus s) sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

le cp1L

pg2

(α minus δ)cp2e

minus δp(tminus s) suprisin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(83)

so

suprisin[st]

eδp(rminus s)

E Ψα v1 s( 1113857(r) minus Ψα v2 s( 1113857(r)

p

Hle cp1

Lpg2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H (84)

where Lg2is the Lipschitz constant of g2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 andslt t

Moreover using (A4) we can show that

suprisin[st]

eδp(rminus s)

E Γα(v s)(r)

p

Hle cp1

Mp

f2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(85)

suprisin[st]

eδp(rminus s)

E Ψα(v s)(r)

p

Hle cp1

Mpg2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(86)

where Mf2andMg2

are the linear growth constants off2 andg2 and cp1 and cp2 are two suitable positive constantsindependent of αgt 0 and slt t

For any fixed adapted process v let us introduce theproblem

dρ(t) A2(t) minus α( 1113857ρ(t)dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) ρ(s) 0

(87)

We denote that its unique mild solution is ρα(v s) ismeans that ρα(v s) solves the equation

ρα(v s)(t) ψα2 ρα(v s) s( 1113857(t) + Γα(v s)(t)

+ Ψα(v s)(t) slt tltT(88)

Due to (82) and (84) using the same argument as (58) in[25] it is easy to prove that for any process v1 v2 and0lt δ lt α we have

suprisin[st]

eδp(rminus s)

E ρα v1 s( 1113857(r) minus ρα v2 s( 1113857(r)

p

H

le cp1Lp

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(89)

where L max Lb2 Lf2

Lg21113966 1113967

Similarly thanks to (85) and (86) for any process v and0lt δ lt α we can prove that

suprisin[st]

eδp(rminus s)

E ρα(v s)(r)

p

Hle cp1

Mp

(α minus δ)cp2

suprisin[st]

eδp(rminus s)

E 1 +v(r)p

H1113872 1113873

(90)

where M max Mb2 Mf2

Mg21113966 1113967

Complexity 11

Lemma 4 Under (A1)ndash(A4) there exists δ gt 0 such that forany x y isin H and pge 1

E vx(t s y)

p

Hle cp 1 +x

p

H + eminus δp(tminus s)

yp

H1113872 1113873 slt t

(91)

Proof We set Λα(t) ≔ vx(t s y) minus ρα(t) whereρα(t) ρα(vx(middot s y) s)(t) is the solution of problem (87)with v vx(middot s y) Using Youngrsquos inequality we have1p

ddtΛα(t)

p

Hlelang A2(t) minus α( 1113857Λα(t)Λα(t)rangH Λα(t)

pminus 2H

+langB2 t xΛα(t) + ρα(t)( 1113857 minus B2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2H

+langB2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2Hle minus α Λα(t)

p

H

+ c Λα(t)

p

H+ c 1 +xH + ρα(t)

H1113872 1113873 Λα(t)

pminus 1H

le minus α Λα(t)

p

H+ cp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(92)

Because α is large enough we can find η α minus cp gt 0such that

ddtΛα(t)

p

Hle minus ηp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(93)

According to the Gronwall inequality we have

Λα(t)

p

Hle e

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(94)

So for any pge 1

vx(t s y)

p

Hle cp ρα(t)

p

H+ cpe

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(95)

Fixing 0lt δ lt η according to (90) we get

eδp(tminus s)

E vx(t s y)

p

Hle cpe

δp(tminus s)E ρα(t)

p

H+ cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873 + cp 1113946t

seδp(rminus s)

E ρα(r)

p

Hdr

le cp1Mp

(α minus δ)cp2sup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873 + cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ cp1Mp

(α minus δ)cp21113946

t

ssup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873dr

(96)

Taking α1 (2cp1Mp)1cp2 + δ when αge α1 we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ 1113946t

ssup

risin[st]

eδp(rminus s)

E vx(r s y)

p

Hdr

(97)

Due to the Gronwall lemma we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

(98)

Hence we get (91)

Lemma 5 Under (A1)ndash(A4) for any t isin R and x y isin H forall pge 1 there exists ηx(t) isin Lp(ΩH) such that

lims⟶minus infin

E vx(t s y) minus ηx

(t)

p

H 0 (99)

Moreover for any pge 1 there exists some δp gt 0 suchthat

E vx(t s y) minus ηx

(t)

p

Hle cpe

minus δp(tminus s) 1 +xp

H +yp

H1113872 1113873

(100)

Finally ηx is a mild solution in R of (79)

Proof Fix hgt 0 and define

ρ(t) vx(t s y) minus v

x(t s minus h y) slt t (101)

We know that ρ(t) is the unique mild solution of theproblem

12 Complexity

dρ(t) A2(t) minus α( 1113857ρ(t) + B2 t x vx(t s y)( 1113857 minus B2 t x vx(t s minus h y)( 11138571113858 1113859dt

+ F2 t x vx(t s y)( 1113857 minus F2 t x vx(t s minus h y)( 11138571113858 1113859dwQ2(t)

+1113946Z

G2 t x vx(t s y)( 1113857 minus G2 t x vx(t s minus h y)( 11138571113858 1113859 1113957N2prime(dt dz)

ρ(s) y minus vx(s s minus h y)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(t) Uα2(t s) y minus vx(s s minus h y)( 1113857 + ψα2(ρ s)(t)

+ 1113946t

sUα2(t r) B2 r x v

x(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

+ 1113946t

sUα2(t r) F2 r x v

x(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r) G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime

(dr dz)

(102)

Multiply both sides of the above equation by eδp(tminus s)Because α large enough according to Lemma 24 in [25] wehave

eδp(tminus s)

Eρ(t)p

H le cpe(δminus α)p(tminus s)

E ec2(ts)A2 y minus v

x(s s minus h y)( 1113857

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

B2 r x vx(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

p

H

+ cpE 1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime(dr dz)

p

H

≔ 11139444

i1I

it

(103)

According to Lemma 31 in [12] we know that for anyJ isinL(Linfin(D)H)capL(H L1(D)) with J Jlowast and sge 0 wehave

esAi JQi

22 leKis

minus βi ρi minus 2( )ρi eminus α ρi+2( )ρi( )s

J2L Linfin(D)H( )

(104)

where

Ki βi

e1113888 1113889

βi ρi minus 2( )ρi

ζ ρi minus 2( )ρi

i κ2ρi

i (105)

Taking pgt 1 such that pβ2(ρ2 minus 2)[ρ2(p minus 2)]lt 1en using the BurkholderndashDavisndashGundy inequality andKunitarsquos first inequality we can get that for any pgep and0lt δ lt α it yields

I2t le cpL

p

b2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se

(δminus α)(tminus r)dr1113888 1113889

p

le cp

Lp

b2

(α minus δ)p suprisin[st]

eδp(rminus s)

Eρ(r)p

H (106)

Complexity 13

I3t le cpE 1113946

t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859Q2

2

2dr1113888 1113889

p2

le cpLp

f2K

p22 1113946

t

se2(δminus α)(tminus r)

c2(t r)minus β2 ρ2minus 2( )ρ2( )e

minus α ρ2+2( )ρ2c2(tr)dr1113888 1113889

p2

times suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se2(δminus α)(tminus r)

c0(t minus r)1113858 1113859minus β2 ρ2minus 2( )ρ2( )dr1113888 1113889

p2

le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0r

minus β2 ρ2minus 2( )ρ2( )(p(pminus 2))dr1113888 1113889

(pminus 2)2

1113946tminus s

0e

minus p(αminus δ)rdr1113888 1113889

le cp

Lp

f2

α minus δsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

I4t le cpE 1113946

t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

p

Hv2prime

(dz)dr

le cpLpg2

suprisin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0e

minus 2(αminus δ)rdr1113888 1113889

p2

+ 1113946tminus s

0e

minus p(αminus δ)rdr⎡⎣ ⎤⎦le cp

Lpg2

(α minus δ)cpsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(107)

Hence we have

suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H

+ cp1Lp

(α minus δ)cp2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(108)

erefore for αgt 0 large enough we can find 0lt δp lt αsuch that

cp1L

α minus δp1113872 1113873cp2lt 1 (109)

is implies that

suprisin[st]

epδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H (110)

Let δp pδp thanks to Lemma 4 we have

E vx(t s y) minus v

x(t s minus h y)

p

H

le cpeminus δp(tminus s)

y minus vx(s s minus h y)

p

H

le cpeminus δp(tminus s) 1 +x

p

H +yp

H + eminus δph

yp

H1113872 1113873

(111)

Because Lp(ΩH) has completeness for any pgep lets⟶ minus infin in (111) there exists ηx(t) isin Lp(ΩH) such that(99) holds en if we let h⟶infin in (111) we obtain (100)Using the Holder inequality we can get (99) and (100) holdsfor any pltp

If we take y1 y2 isin H use the same arguments forvx(t s y1) minus vx(t s y2) slt t we have

E vx

t s y1( 1113857 minus vx

t s y2( 1113857

p

Hle cpe

minus δp(tminus s)y1 minus y2

p

H slt t

(112)

Let s⟶ minus infin this means that the limit ηx(t) does notdepend on the initial condition y isin H

Finally we prove that ηx(t) is a mild solution of (79)Due to the limit ηx(t) does not depend on the initialcondition we can let initial condition y 0 For any slt t

and hgt 0 we have

vx(t s minus h 0) Uα2(t s)v

x(s s minus h 0) + ψα2 v

x(middot s minus h 0) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s minus h 0)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s minus h 0)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s minus h 0) z( 1113857 1113957N2prime

(dr dz)

(113)

Let h⟶infin on both sides due to (99) we can get forany slt t have

ηx(t) Uα2(t s)ηx

(s) + ψα2 ηx s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x ηx

(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x ηx

(r)( 1113857dwQ2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x ηx(r) z( 1113857 1113957N2prime

(dr dz)

(114)

is means that ηx(t) is a mild solution of (79)For any t isin R and x isin H we denote that the law of the

random variable ηx(t) is μxt and we introduce the transition

evolution operator

Pxstφ(y) Eφ v

x(t s y)( 1113857 slt t y isin H (115)

14 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

E uϵ(t + h) minus uϵ(t)

p

H

le cpT h(1minus 1p )p

+ hp

+ h(12)minus (1p)minus β1 ρ1minus 2( )2ρ1( )p

+ hp2

+ h1113874 1113875

times 1 +xp

H +yp

H1113872 1113873 + cpθThθp 1 +x

p

θ +yp

H1113872 1113873

(73)

As we are assuming |h|le 1 (68) follows for any pgep bytaking

β(θ) ≔ min θ 1 minus1p

112

minus1p

minusβ1 ρ1 minus 2( 1113857

2ρ112

1113896 1113897 (74)

From the Holder inequality we can estimate (68) forpltp

E uϵ(t + h) minus uϵ(t)

p

Hle E uϵ(t + h) minus uϵ(t)

p

H1113876 1113877

pp

(75)

So we have (68)According to the above lemma we get that for everyϵ isin (0 1] the function uϵ(t) is uniformly bounded aboutt isin [0 T] and it is also equicontinuous at every point oft isin [0 T] In view of eorem 123 in [43] we can infer thatthe set uϵ1113864 1113865ϵisin(01] is relatively compact in D([0 T]H) Inaddition according to eorem 132 in [43] and the abovelemma by using Chebyshevrsquos inequality we also can get thatthe family of probability measures L(uϵ)1113864 1113865ϵisin(01] is tight inP(D([0 T]H))

4 An Evolution Family of Measures for theFast Equation

For any frozen slow component x isin H any initial conditiony isin H and any s isin R we introduce the following problem

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) v(s) y

(76)

where

wQ2(t)

wQ21 (t)

wQ22 (minus t)

⎧⎨

⎩

if tge 0

if tlt 0

1113957N2prime(t z)

1113957N1prime(t z)

1113957N3prime(minus t z)

⎧⎪⎪⎨

⎪⎪⎩

if tge 0

if tlt 0

(77)

for two independent Q2-Wiener processes wQ21 (t) and

wQ22 (t) and two independent compensated Poisson mea-

sures 1113957N1prime(dt dz) and 1113957N3prime(dt dz) with the same Levymeasure are both defined as in Section 2

According to the definition of the operator ψα2(middot s) weknow that the mapping ψα2(middot s) C([s T] H)⟶C([s T]H) is a linearly bounded operator and it is Lip-schitz continuous Hence we have that for anyx y isin H pge 1 and sltT there exists a unique mild solution

vx(middot s y) isin Lp(Ω D([0 T]H)) (see [28]) in the followingform

vx(t s y) Uα2(t s)y + ψα2 v

x(middot s y) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s y)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s y)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s y) z( 1113857 1113957N2prime(dr dz)

(78)

Moreover if the space D(RH) endowed with the to-pology of uniform convergence on bounded intervals anFt1113864 1113865tisinR-adapted process vx isin Lp(Ω D([0 T]H)) is a mildsolution of the equation

dv(t) A2(t) minus α( 1113857v(t) + B2(t x v(t))1113858 1113859dt

+ F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz)

(79)

where t isin R en for every slt t we have

vx(t) Uα2(t s)v

x(s) + ψα2 v

x s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r) z( 1113857 1113957N2prime

(dr dz)

(80)

In the following for any x isin H and any adapted processv we set

Γα(v s)(t) ≔ 1113946t

sUα2(t r)F2(r x v(r))dw

Q2(r) tgt s

Ψα(v s)(t) ≔ 1113946t

s1113946Z

Uα2(t r)G2(r x v(r) z) 1113957N2prime(dr dz) tgt s

(81)

For any 0lt δ lt α and any v1 and v2 with slt t by pro-ceeding as in the proof of Lemma 71 in [44] it is possible toshow that there exists pgt 1 such that for any pgep we have

suprisin[st]

eδp(rminus s)

E Γα v1 s( 1113857(r) minus Γα v2 s( 1113857(r)

p

H

le cp1L

p

f2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(82)

where Lf2is the Lipschitz constant of f2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 and slt tFor the stochastic term Ψα(v s)(t) using Kunitarsquos first

inequality ([26] eorem 4423) we get

10 Complexity

E Ψα v1 s( 1113857(t) minus Ψα v2 s( 1113857(t)

p

Hle cpE 1113946

t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

p

Hv2prime(dz)dr

le cpE 1113946t

s1113946Z

eminus 2α(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

2H

v2prime(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus αp(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

p

Hv2prime(dz)dr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)e

minus 2δ(tminus s)e2δ(rminus s)

E v1(r) minus v2(r)

2Hdr1113888 1113889

p2

+ cpLpg2

1113946t

se

minus p(αminus δ)(tminus r)e

minus δp(tminus s)eδp(rminus s)

E v1(r) minus v2(r)

p

Hdr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)dr1113888 1113889

p2

+ 1113946t

se

minus p(αminus δ)(tminus r)dr⎡⎣ ⎤⎦ times eminus δp(tminus s) sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

le cp1L

pg2

(α minus δ)cp2e

minus δp(tminus s) suprisin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(83)

so

suprisin[st]

eδp(rminus s)

E Ψα v1 s( 1113857(r) minus Ψα v2 s( 1113857(r)

p

Hle cp1

Lpg2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H (84)

where Lg2is the Lipschitz constant of g2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 andslt t

Moreover using (A4) we can show that

suprisin[st]

eδp(rminus s)

E Γα(v s)(r)

p

Hle cp1

Mp

f2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(85)

suprisin[st]

eδp(rminus s)

E Ψα(v s)(r)

p

Hle cp1

Mpg2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(86)

where Mf2andMg2

are the linear growth constants off2 andg2 and cp1 and cp2 are two suitable positive constantsindependent of αgt 0 and slt t

For any fixed adapted process v let us introduce theproblem

dρ(t) A2(t) minus α( 1113857ρ(t)dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) ρ(s) 0

(87)

We denote that its unique mild solution is ρα(v s) ismeans that ρα(v s) solves the equation

ρα(v s)(t) ψα2 ρα(v s) s( 1113857(t) + Γα(v s)(t)

+ Ψα(v s)(t) slt tltT(88)

Due to (82) and (84) using the same argument as (58) in[25] it is easy to prove that for any process v1 v2 and0lt δ lt α we have

suprisin[st]

eδp(rminus s)

E ρα v1 s( 1113857(r) minus ρα v2 s( 1113857(r)

p

H

le cp1Lp

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(89)

where L max Lb2 Lf2

Lg21113966 1113967

Similarly thanks to (85) and (86) for any process v and0lt δ lt α we can prove that

suprisin[st]

eδp(rminus s)

E ρα(v s)(r)

p

Hle cp1

Mp

(α minus δ)cp2

suprisin[st]

eδp(rminus s)

E 1 +v(r)p

H1113872 1113873

(90)

where M max Mb2 Mf2

Mg21113966 1113967

Complexity 11

Lemma 4 Under (A1)ndash(A4) there exists δ gt 0 such that forany x y isin H and pge 1

E vx(t s y)

p

Hle cp 1 +x

p

H + eminus δp(tminus s)

yp

H1113872 1113873 slt t

(91)

Proof We set Λα(t) ≔ vx(t s y) minus ρα(t) whereρα(t) ρα(vx(middot s y) s)(t) is the solution of problem (87)with v vx(middot s y) Using Youngrsquos inequality we have1p

ddtΛα(t)

p

Hlelang A2(t) minus α( 1113857Λα(t)Λα(t)rangH Λα(t)

pminus 2H

+langB2 t xΛα(t) + ρα(t)( 1113857 minus B2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2H

+langB2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2Hle minus α Λα(t)

p

H

+ c Λα(t)

p

H+ c 1 +xH + ρα(t)

H1113872 1113873 Λα(t)

pminus 1H

le minus α Λα(t)

p

H+ cp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(92)

Because α is large enough we can find η α minus cp gt 0such that

ddtΛα(t)

p

Hle minus ηp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(93)

According to the Gronwall inequality we have

Λα(t)

p

Hle e

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(94)

So for any pge 1

vx(t s y)

p

Hle cp ρα(t)

p

H+ cpe

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(95)

Fixing 0lt δ lt η according to (90) we get

eδp(tminus s)

E vx(t s y)

p

Hle cpe

δp(tminus s)E ρα(t)

p

H+ cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873 + cp 1113946t

seδp(rminus s)

E ρα(r)

p

Hdr

le cp1Mp

(α minus δ)cp2sup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873 + cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ cp1Mp

(α minus δ)cp21113946

t

ssup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873dr

(96)

Taking α1 (2cp1Mp)1cp2 + δ when αge α1 we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ 1113946t

ssup

risin[st]

eδp(rminus s)

E vx(r s y)

p

Hdr

(97)

Due to the Gronwall lemma we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

(98)

Hence we get (91)

Lemma 5 Under (A1)ndash(A4) for any t isin R and x y isin H forall pge 1 there exists ηx(t) isin Lp(ΩH) such that

lims⟶minus infin

E vx(t s y) minus ηx

(t)

p

H 0 (99)

Moreover for any pge 1 there exists some δp gt 0 suchthat

E vx(t s y) minus ηx

(t)

p

Hle cpe

minus δp(tminus s) 1 +xp

H +yp

H1113872 1113873

(100)

Finally ηx is a mild solution in R of (79)

Proof Fix hgt 0 and define

ρ(t) vx(t s y) minus v

x(t s minus h y) slt t (101)

We know that ρ(t) is the unique mild solution of theproblem

12 Complexity

dρ(t) A2(t) minus α( 1113857ρ(t) + B2 t x vx(t s y)( 1113857 minus B2 t x vx(t s minus h y)( 11138571113858 1113859dt

+ F2 t x vx(t s y)( 1113857 minus F2 t x vx(t s minus h y)( 11138571113858 1113859dwQ2(t)

+1113946Z

G2 t x vx(t s y)( 1113857 minus G2 t x vx(t s minus h y)( 11138571113858 1113859 1113957N2prime(dt dz)

ρ(s) y minus vx(s s minus h y)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(t) Uα2(t s) y minus vx(s s minus h y)( 1113857 + ψα2(ρ s)(t)

+ 1113946t

sUα2(t r) B2 r x v

x(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

+ 1113946t

sUα2(t r) F2 r x v

x(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r) G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime

(dr dz)

(102)

Multiply both sides of the above equation by eδp(tminus s)Because α large enough according to Lemma 24 in [25] wehave

eδp(tminus s)

Eρ(t)p

H le cpe(δminus α)p(tminus s)

E ec2(ts)A2 y minus v

x(s s minus h y)( 1113857

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

B2 r x vx(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

p

H

+ cpE 1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime(dr dz)

p

H

≔ 11139444

i1I

it

(103)

According to Lemma 31 in [12] we know that for anyJ isinL(Linfin(D)H)capL(H L1(D)) with J Jlowast and sge 0 wehave

esAi JQi

22 leKis

minus βi ρi minus 2( )ρi eminus α ρi+2( )ρi( )s

J2L Linfin(D)H( )

(104)

where

Ki βi

e1113888 1113889

βi ρi minus 2( )ρi

ζ ρi minus 2( )ρi

i κ2ρi

i (105)

Taking pgt 1 such that pβ2(ρ2 minus 2)[ρ2(p minus 2)]lt 1en using the BurkholderndashDavisndashGundy inequality andKunitarsquos first inequality we can get that for any pgep and0lt δ lt α it yields

I2t le cpL

p

b2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se

(δminus α)(tminus r)dr1113888 1113889

p

le cp

Lp

b2

(α minus δ)p suprisin[st]

eδp(rminus s)

Eρ(r)p

H (106)

Complexity 13

I3t le cpE 1113946

t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859Q2

2

2dr1113888 1113889

p2

le cpLp

f2K

p22 1113946

t

se2(δminus α)(tminus r)

c2(t r)minus β2 ρ2minus 2( )ρ2( )e

minus α ρ2+2( )ρ2c2(tr)dr1113888 1113889

p2

times suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se2(δminus α)(tminus r)

c0(t minus r)1113858 1113859minus β2 ρ2minus 2( )ρ2( )dr1113888 1113889

p2

le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0r

minus β2 ρ2minus 2( )ρ2( )(p(pminus 2))dr1113888 1113889

(pminus 2)2

1113946tminus s

0e

minus p(αminus δ)rdr1113888 1113889

le cp

Lp

f2

α minus δsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

I4t le cpE 1113946

t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

p

Hv2prime

(dz)dr

le cpLpg2

suprisin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0e

minus 2(αminus δ)rdr1113888 1113889

p2

+ 1113946tminus s

0e

minus p(αminus δ)rdr⎡⎣ ⎤⎦le cp

Lpg2

(α minus δ)cpsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(107)

Hence we have

suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H

+ cp1Lp

(α minus δ)cp2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(108)

erefore for αgt 0 large enough we can find 0lt δp lt αsuch that

cp1L

α minus δp1113872 1113873cp2lt 1 (109)

is implies that

suprisin[st]

epδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H (110)

Let δp pδp thanks to Lemma 4 we have

E vx(t s y) minus v

x(t s minus h y)

p

H

le cpeminus δp(tminus s)

y minus vx(s s minus h y)

p

H

le cpeminus δp(tminus s) 1 +x

p

H +yp

H + eminus δph

yp

H1113872 1113873

(111)

Because Lp(ΩH) has completeness for any pgep lets⟶ minus infin in (111) there exists ηx(t) isin Lp(ΩH) such that(99) holds en if we let h⟶infin in (111) we obtain (100)Using the Holder inequality we can get (99) and (100) holdsfor any pltp

If we take y1 y2 isin H use the same arguments forvx(t s y1) minus vx(t s y2) slt t we have

E vx

t s y1( 1113857 minus vx

t s y2( 1113857

p

Hle cpe

minus δp(tminus s)y1 minus y2

p

H slt t

(112)

Let s⟶ minus infin this means that the limit ηx(t) does notdepend on the initial condition y isin H

Finally we prove that ηx(t) is a mild solution of (79)Due to the limit ηx(t) does not depend on the initialcondition we can let initial condition y 0 For any slt t

and hgt 0 we have

vx(t s minus h 0) Uα2(t s)v

x(s s minus h 0) + ψα2 v

x(middot s minus h 0) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s minus h 0)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s minus h 0)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s minus h 0) z( 1113857 1113957N2prime

(dr dz)

(113)

Let h⟶infin on both sides due to (99) we can get forany slt t have

ηx(t) Uα2(t s)ηx

(s) + ψα2 ηx s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x ηx

(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x ηx

(r)( 1113857dwQ2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x ηx(r) z( 1113857 1113957N2prime

(dr dz)

(114)

is means that ηx(t) is a mild solution of (79)For any t isin R and x isin H we denote that the law of the

random variable ηx(t) is μxt and we introduce the transition

evolution operator

Pxstφ(y) Eφ v

x(t s y)( 1113857 slt t y isin H (115)

14 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

E Ψα v1 s( 1113857(t) minus Ψα v2 s( 1113857(t)

p

Hle cpE 1113946

t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus α(tminus r)

ec2(tr)A2 G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 11138571113858 1113859

p

Hv2prime(dz)dr

le cpE 1113946t

s1113946Z

eminus 2α(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

2H

v2prime(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

eminus αp(tminus r)

G2 r x v1(r) z( 1113857 minus G2 r x v2(r) z( 1113857

p

Hv2prime(dz)dr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)e

minus 2δ(tminus s)e2δ(rminus s)

E v1(r) minus v2(r)

2Hdr1113888 1113889

p2

+ cpLpg2

1113946t

se

minus p(αminus δ)(tminus r)e

minus δp(tminus s)eδp(rminus s)

E v1(r) minus v2(r)

p

Hdr

le cpLpg2

1113946t

se

minus 2(αminus δ)(tminus r)dr1113888 1113889

p2

+ 1113946t

se

minus p(αminus δ)(tminus r)dr⎡⎣ ⎤⎦ times eminus δp(tminus s) sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

le cp1L

pg2

(α minus δ)cp2e

minus δp(tminus s) suprisin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(83)

so

suprisin[st]

eδp(rminus s)

E Ψα v1 s( 1113857(r) minus Ψα v2 s( 1113857(r)

p

Hle cp1

Lpg2

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H (84)

where Lg2is the Lipschitz constant of g2 and cp1 and cp2 are

two suitable positive constants independent of αgt 0 andslt t

Moreover using (A4) we can show that

suprisin[st]

eδp(rminus s)

E Γα(v s)(r)

p

Hle cp1

Mp

f2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(85)

suprisin[st]

eδp(rminus s)

E Ψα(v s)(r)

p

Hle cp1

Mpg2

(α minus δ)cp2

suprisin[st]

eδp(rminus s) 1 + Ev(r)

p

H1113872 1113873

(86)

where Mf2andMg2

are the linear growth constants off2 andg2 and cp1 and cp2 are two suitable positive constantsindependent of αgt 0 and slt t

For any fixed adapted process v let us introduce theproblem

dρ(t) A2(t) minus α( 1113857ρ(t)dt + F2(t x v(t))dωQ2(t)

+ 1113946Z

G2(t x v(t) z) 1113957N2prime(dt dz) ρ(s) 0

(87)

We denote that its unique mild solution is ρα(v s) ismeans that ρα(v s) solves the equation

ρα(v s)(t) ψα2 ρα(v s) s( 1113857(t) + Γα(v s)(t)

+ Ψα(v s)(t) slt tltT(88)

Due to (82) and (84) using the same argument as (58) in[25] it is easy to prove that for any process v1 v2 and0lt δ lt α we have

suprisin[st]

eδp(rminus s)

E ρα v1 s( 1113857(r) minus ρα v2 s( 1113857(r)

p

H

le cp1Lp

(α minus δ)cp2times sup

risin[st]

eδp(rminus s)

E v1(r) minus v2(r)

p

H

(89)

where L max Lb2 Lf2

Lg21113966 1113967

Similarly thanks to (85) and (86) for any process v and0lt δ lt α we can prove that

suprisin[st]

eδp(rminus s)

E ρα(v s)(r)

p

Hle cp1

Mp

(α minus δ)cp2

suprisin[st]

eδp(rminus s)

E 1 +v(r)p

H1113872 1113873

(90)

where M max Mb2 Mf2

Mg21113966 1113967

Complexity 11

Lemma 4 Under (A1)ndash(A4) there exists δ gt 0 such that forany x y isin H and pge 1

E vx(t s y)

p

Hle cp 1 +x

p

H + eminus δp(tminus s)

yp

H1113872 1113873 slt t

(91)

Proof We set Λα(t) ≔ vx(t s y) minus ρα(t) whereρα(t) ρα(vx(middot s y) s)(t) is the solution of problem (87)with v vx(middot s y) Using Youngrsquos inequality we have1p

ddtΛα(t)

p

Hlelang A2(t) minus α( 1113857Λα(t)Λα(t)rangH Λα(t)

pminus 2H

+langB2 t xΛα(t) + ρα(t)( 1113857 minus B2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2H

+langB2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2Hle minus α Λα(t)

p

H

+ c Λα(t)

p

H+ c 1 +xH + ρα(t)

H1113872 1113873 Λα(t)

pminus 1H

le minus α Λα(t)

p

H+ cp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(92)

Because α is large enough we can find η α minus cp gt 0such that

ddtΛα(t)

p

Hle minus ηp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(93)

According to the Gronwall inequality we have

Λα(t)

p

Hle e

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(94)

So for any pge 1

vx(t s y)

p

Hle cp ρα(t)

p

H+ cpe

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(95)

Fixing 0lt δ lt η according to (90) we get

eδp(tminus s)

E vx(t s y)

p

Hle cpe

δp(tminus s)E ρα(t)

p

H+ cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873 + cp 1113946t

seδp(rminus s)

E ρα(r)

p

Hdr

le cp1Mp

(α minus δ)cp2sup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873 + cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ cp1Mp

(α minus δ)cp21113946

t

ssup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873dr

(96)

Taking α1 (2cp1Mp)1cp2 + δ when αge α1 we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ 1113946t

ssup

risin[st]

eδp(rminus s)

E vx(r s y)

p

Hdr

(97)

Due to the Gronwall lemma we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

(98)

Hence we get (91)

Lemma 5 Under (A1)ndash(A4) for any t isin R and x y isin H forall pge 1 there exists ηx(t) isin Lp(ΩH) such that

lims⟶minus infin

E vx(t s y) minus ηx

(t)

p

H 0 (99)

Moreover for any pge 1 there exists some δp gt 0 suchthat

E vx(t s y) minus ηx

(t)

p

Hle cpe

minus δp(tminus s) 1 +xp

H +yp

H1113872 1113873

(100)

Finally ηx is a mild solution in R of (79)

Proof Fix hgt 0 and define

ρ(t) vx(t s y) minus v

x(t s minus h y) slt t (101)

We know that ρ(t) is the unique mild solution of theproblem

12 Complexity

dρ(t) A2(t) minus α( 1113857ρ(t) + B2 t x vx(t s y)( 1113857 minus B2 t x vx(t s minus h y)( 11138571113858 1113859dt

+ F2 t x vx(t s y)( 1113857 minus F2 t x vx(t s minus h y)( 11138571113858 1113859dwQ2(t)

+1113946Z

G2 t x vx(t s y)( 1113857 minus G2 t x vx(t s minus h y)( 11138571113858 1113859 1113957N2prime(dt dz)

ρ(s) y minus vx(s s minus h y)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(t) Uα2(t s) y minus vx(s s minus h y)( 1113857 + ψα2(ρ s)(t)

+ 1113946t

sUα2(t r) B2 r x v

x(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

+ 1113946t

sUα2(t r) F2 r x v

x(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r) G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime

(dr dz)

(102)

Multiply both sides of the above equation by eδp(tminus s)Because α large enough according to Lemma 24 in [25] wehave

eδp(tminus s)

Eρ(t)p

H le cpe(δminus α)p(tminus s)

E ec2(ts)A2 y minus v

x(s s minus h y)( 1113857

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

B2 r x vx(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

p

H

+ cpE 1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime(dr dz)

p

H

≔ 11139444

i1I

it

(103)

According to Lemma 31 in [12] we know that for anyJ isinL(Linfin(D)H)capL(H L1(D)) with J Jlowast and sge 0 wehave

esAi JQi

22 leKis

minus βi ρi minus 2( )ρi eminus α ρi+2( )ρi( )s

J2L Linfin(D)H( )

(104)

where

Ki βi

e1113888 1113889

βi ρi minus 2( )ρi

ζ ρi minus 2( )ρi

i κ2ρi

i (105)

Taking pgt 1 such that pβ2(ρ2 minus 2)[ρ2(p minus 2)]lt 1en using the BurkholderndashDavisndashGundy inequality andKunitarsquos first inequality we can get that for any pgep and0lt δ lt α it yields

I2t le cpL

p

b2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se

(δminus α)(tminus r)dr1113888 1113889

p

le cp

Lp

b2

(α minus δ)p suprisin[st]

eδp(rminus s)

Eρ(r)p

H (106)

Complexity 13

I3t le cpE 1113946

t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859Q2

2

2dr1113888 1113889

p2

le cpLp

f2K

p22 1113946

t

se2(δminus α)(tminus r)

c2(t r)minus β2 ρ2minus 2( )ρ2( )e

minus α ρ2+2( )ρ2c2(tr)dr1113888 1113889

p2

times suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se2(δminus α)(tminus r)

c0(t minus r)1113858 1113859minus β2 ρ2minus 2( )ρ2( )dr1113888 1113889

p2

le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0r

minus β2 ρ2minus 2( )ρ2( )(p(pminus 2))dr1113888 1113889

(pminus 2)2

1113946tminus s

0e

minus p(αminus δ)rdr1113888 1113889

le cp

Lp

f2

α minus δsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

I4t le cpE 1113946

t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

p

Hv2prime

(dz)dr

le cpLpg2

suprisin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0e

minus 2(αminus δ)rdr1113888 1113889

p2

+ 1113946tminus s

0e

minus p(αminus δ)rdr⎡⎣ ⎤⎦le cp

Lpg2

(α minus δ)cpsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(107)

Hence we have

suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H

+ cp1Lp

(α minus δ)cp2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(108)

erefore for αgt 0 large enough we can find 0lt δp lt αsuch that

cp1L

α minus δp1113872 1113873cp2lt 1 (109)

is implies that

suprisin[st]

epδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H (110)

Let δp pδp thanks to Lemma 4 we have

E vx(t s y) minus v

x(t s minus h y)

p

H

le cpeminus δp(tminus s)

y minus vx(s s minus h y)

p

H

le cpeminus δp(tminus s) 1 +x

p

H +yp

H + eminus δph

yp

H1113872 1113873

(111)

Because Lp(ΩH) has completeness for any pgep lets⟶ minus infin in (111) there exists ηx(t) isin Lp(ΩH) such that(99) holds en if we let h⟶infin in (111) we obtain (100)Using the Holder inequality we can get (99) and (100) holdsfor any pltp

If we take y1 y2 isin H use the same arguments forvx(t s y1) minus vx(t s y2) slt t we have

E vx

t s y1( 1113857 minus vx

t s y2( 1113857

p

Hle cpe

minus δp(tminus s)y1 minus y2

p

H slt t

(112)

Let s⟶ minus infin this means that the limit ηx(t) does notdepend on the initial condition y isin H

Finally we prove that ηx(t) is a mild solution of (79)Due to the limit ηx(t) does not depend on the initialcondition we can let initial condition y 0 For any slt t

and hgt 0 we have

vx(t s minus h 0) Uα2(t s)v

x(s s minus h 0) + ψα2 v

x(middot s minus h 0) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s minus h 0)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s minus h 0)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s minus h 0) z( 1113857 1113957N2prime

(dr dz)

(113)

Let h⟶infin on both sides due to (99) we can get forany slt t have

ηx(t) Uα2(t s)ηx

(s) + ψα2 ηx s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x ηx

(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x ηx

(r)( 1113857dwQ2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x ηx(r) z( 1113857 1113957N2prime

(dr dz)

(114)

is means that ηx(t) is a mild solution of (79)For any t isin R and x isin H we denote that the law of the

random variable ηx(t) is μxt and we introduce the transition

evolution operator

Pxstφ(y) Eφ v

x(t s y)( 1113857 slt t y isin H (115)

14 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

Lemma 4 Under (A1)ndash(A4) there exists δ gt 0 such that forany x y isin H and pge 1

E vx(t s y)

p

Hle cp 1 +x

p

H + eminus δp(tminus s)

yp

H1113872 1113873 slt t

(91)

Proof We set Λα(t) ≔ vx(t s y) minus ρα(t) whereρα(t) ρα(vx(middot s y) s)(t) is the solution of problem (87)with v vx(middot s y) Using Youngrsquos inequality we have1p

ddtΛα(t)

p

Hlelang A2(t) minus α( 1113857Λα(t)Λα(t)rangH Λα(t)

pminus 2H

+langB2 t xΛα(t) + ρα(t)( 1113857 minus B2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2H

+langB2 t x ρα(t)( 1113857Λα(t)rangH Λα(t)

pminus 2Hle minus α Λα(t)

p

H

+ c Λα(t)

p

H+ c 1 +xH + ρα(t)

H1113872 1113873 Λα(t)

pminus 1H

le minus α Λα(t)

p

H+ cp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(92)

Because α is large enough we can find η α minus cp gt 0such that

ddtΛα(t)

p

Hle minus ηp Λα(t)

p

H+ cp 1 +x

p

H + ρα(t)

p

H1113872 1113873

(93)

According to the Gronwall inequality we have

Λα(t)

p

Hle e

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(94)

So for any pge 1

vx(t s y)

p

Hle cp ρα(t)

p

H+ cpe

minus ηp(tminus s)y

p

H + cp 1 +xp

H1113872 1113873

+ cp 1113946t

se

minus ηp(tminus r) ρα(r)

p

Hdr

(95)

Fixing 0lt δ lt η according to (90) we get

eδp(tminus s)

E vx(t s y)

p

Hle cpe

δp(tminus s)E ρα(t)

p

H+ cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873 + cp 1113946t

seδp(rminus s)

E ρα(r)

p

Hdr

le cp1Mp

(α minus δ)cp2sup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873 + cpe

p(δminus η)(tminus s)y

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ cp1Mp

(α minus δ)cp21113946

t

ssup

risin[st]

eδp(rminus s) 1 + E v

x(r s y)

p

H1113872 1113873dr

(96)

Taking α1 (2cp1Mp)1cp2 + δ when αge α1 we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

+ 1113946t

ssup

risin[st]

eδp(rminus s)

E vx(r s y)

p

Hdr

(97)

Due to the Gronwall lemma we have

suprisin[st]

eδp(rminus s)

E vx(r s y)

p

Hle cpy

p

H + cpeδp(tminus s) 1 +x

p

H1113872 1113873

(98)

Hence we get (91)

Lemma 5 Under (A1)ndash(A4) for any t isin R and x y isin H forall pge 1 there exists ηx(t) isin Lp(ΩH) such that

lims⟶minus infin

E vx(t s y) minus ηx

(t)

p

H 0 (99)

Moreover for any pge 1 there exists some δp gt 0 suchthat

E vx(t s y) minus ηx

(t)

p

Hle cpe

minus δp(tminus s) 1 +xp

H +yp

H1113872 1113873

(100)

Finally ηx is a mild solution in R of (79)

Proof Fix hgt 0 and define

ρ(t) vx(t s y) minus v

x(t s minus h y) slt t (101)

We know that ρ(t) is the unique mild solution of theproblem

12 Complexity

dρ(t) A2(t) minus α( 1113857ρ(t) + B2 t x vx(t s y)( 1113857 minus B2 t x vx(t s minus h y)( 11138571113858 1113859dt

+ F2 t x vx(t s y)( 1113857 minus F2 t x vx(t s minus h y)( 11138571113858 1113859dwQ2(t)

+1113946Z

G2 t x vx(t s y)( 1113857 minus G2 t x vx(t s minus h y)( 11138571113858 1113859 1113957N2prime(dt dz)

ρ(s) y minus vx(s s minus h y)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(t) Uα2(t s) y minus vx(s s minus h y)( 1113857 + ψα2(ρ s)(t)

+ 1113946t

sUα2(t r) B2 r x v

x(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

+ 1113946t

sUα2(t r) F2 r x v

x(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r) G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime

(dr dz)

(102)

Multiply both sides of the above equation by eδp(tminus s)Because α large enough according to Lemma 24 in [25] wehave

eδp(tminus s)

Eρ(t)p

H le cpe(δminus α)p(tminus s)

E ec2(ts)A2 y minus v

x(s s minus h y)( 1113857

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

B2 r x vx(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

p

H

+ cpE 1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime(dr dz)

p

H

≔ 11139444

i1I

it

(103)

According to Lemma 31 in [12] we know that for anyJ isinL(Linfin(D)H)capL(H L1(D)) with J Jlowast and sge 0 wehave

esAi JQi

22 leKis

minus βi ρi minus 2( )ρi eminus α ρi+2( )ρi( )s

J2L Linfin(D)H( )

(104)

where

Ki βi

e1113888 1113889

βi ρi minus 2( )ρi

ζ ρi minus 2( )ρi

i κ2ρi

i (105)

Taking pgt 1 such that pβ2(ρ2 minus 2)[ρ2(p minus 2)]lt 1en using the BurkholderndashDavisndashGundy inequality andKunitarsquos first inequality we can get that for any pgep and0lt δ lt α it yields

I2t le cpL

p

b2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se

(δminus α)(tminus r)dr1113888 1113889

p

le cp

Lp

b2

(α minus δ)p suprisin[st]

eδp(rminus s)

Eρ(r)p

H (106)

Complexity 13

I3t le cpE 1113946

t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859Q2

2

2dr1113888 1113889

p2

le cpLp

f2K

p22 1113946

t

se2(δminus α)(tminus r)

c2(t r)minus β2 ρ2minus 2( )ρ2( )e

minus α ρ2+2( )ρ2c2(tr)dr1113888 1113889

p2

times suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se2(δminus α)(tminus r)

c0(t minus r)1113858 1113859minus β2 ρ2minus 2( )ρ2( )dr1113888 1113889

p2

le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0r

minus β2 ρ2minus 2( )ρ2( )(p(pminus 2))dr1113888 1113889

(pminus 2)2

1113946tminus s

0e

minus p(αminus δ)rdr1113888 1113889

le cp

Lp

f2

α minus δsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

I4t le cpE 1113946

t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

p

Hv2prime

(dz)dr

le cpLpg2

suprisin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0e

minus 2(αminus δ)rdr1113888 1113889

p2

+ 1113946tminus s

0e

minus p(αminus δ)rdr⎡⎣ ⎤⎦le cp

Lpg2

(α minus δ)cpsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(107)

Hence we have

suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H

+ cp1Lp

(α minus δ)cp2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(108)

erefore for αgt 0 large enough we can find 0lt δp lt αsuch that

cp1L

α minus δp1113872 1113873cp2lt 1 (109)

is implies that

suprisin[st]

epδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H (110)

Let δp pδp thanks to Lemma 4 we have

E vx(t s y) minus v

x(t s minus h y)

p

H

le cpeminus δp(tminus s)

y minus vx(s s minus h y)

p

H

le cpeminus δp(tminus s) 1 +x

p

H +yp

H + eminus δph

yp

H1113872 1113873

(111)

Because Lp(ΩH) has completeness for any pgep lets⟶ minus infin in (111) there exists ηx(t) isin Lp(ΩH) such that(99) holds en if we let h⟶infin in (111) we obtain (100)Using the Holder inequality we can get (99) and (100) holdsfor any pltp

If we take y1 y2 isin H use the same arguments forvx(t s y1) minus vx(t s y2) slt t we have

E vx

t s y1( 1113857 minus vx

t s y2( 1113857

p

Hle cpe

minus δp(tminus s)y1 minus y2

p

H slt t

(112)

Let s⟶ minus infin this means that the limit ηx(t) does notdepend on the initial condition y isin H

Finally we prove that ηx(t) is a mild solution of (79)Due to the limit ηx(t) does not depend on the initialcondition we can let initial condition y 0 For any slt t

and hgt 0 we have

vx(t s minus h 0) Uα2(t s)v

x(s s minus h 0) + ψα2 v

x(middot s minus h 0) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s minus h 0)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s minus h 0)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s minus h 0) z( 1113857 1113957N2prime

(dr dz)

(113)

Let h⟶infin on both sides due to (99) we can get forany slt t have

ηx(t) Uα2(t s)ηx

(s) + ψα2 ηx s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x ηx

(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x ηx

(r)( 1113857dwQ2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x ηx(r) z( 1113857 1113957N2prime

(dr dz)

(114)

is means that ηx(t) is a mild solution of (79)For any t isin R and x isin H we denote that the law of the

random variable ηx(t) is μxt and we introduce the transition

evolution operator

Pxstφ(y) Eφ v

x(t s y)( 1113857 slt t y isin H (115)

14 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

dρ(t) A2(t) minus α( 1113857ρ(t) + B2 t x vx(t s y)( 1113857 minus B2 t x vx(t s minus h y)( 11138571113858 1113859dt

+ F2 t x vx(t s y)( 1113857 minus F2 t x vx(t s minus h y)( 11138571113858 1113859dwQ2(t)

+1113946Z

G2 t x vx(t s y)( 1113857 minus G2 t x vx(t s minus h y)( 11138571113858 1113859 1113957N2prime(dt dz)

ρ(s) y minus vx(s s minus h y)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ(t) Uα2(t s) y minus vx(s s minus h y)( 1113857 + ψα2(ρ s)(t)

+ 1113946t

sUα2(t r) B2 r x v

x(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

+ 1113946t

sUα2(t r) F2 r x v

x(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r) G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime

(dr dz)

(102)

Multiply both sides of the above equation by eδp(tminus s)Because α large enough according to Lemma 24 in [25] wehave

eδp(tminus s)

Eρ(t)p

H le cpe(δminus α)p(tminus s)

E ec2(ts)A2 y minus v

x(s s minus h y)( 1113857

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

B2 r x vx(r s y)( 1113857 minus B2 r x v

x(r s minus h y)( 11138571113858 1113859dr

p

H

+ cpE 1113946t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859dw

Q2(r)

p

H

+ cpE 1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859 1113957N2prime(dr dz)

p

H

≔ 11139444

i1I

it

(103)

According to Lemma 31 in [12] we know that for anyJ isinL(Linfin(D)H)capL(H L1(D)) with J Jlowast and sge 0 wehave

esAi JQi

22 leKis

minus βi ρi minus 2( )ρi eminus α ρi+2( )ρi( )s

J2L Linfin(D)H( )

(104)

where

Ki βi

e1113888 1113889

βi ρi minus 2( )ρi

ζ ρi minus 2( )ρi

i κ2ρi

i (105)

Taking pgt 1 such that pβ2(ρ2 minus 2)[ρ2(p minus 2)]lt 1en using the BurkholderndashDavisndashGundy inequality andKunitarsquos first inequality we can get that for any pgep and0lt δ lt α it yields

I2t le cpL

p

b2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se

(δminus α)(tminus r)dr1113888 1113889

p

le cp

Lp

b2

(α minus δ)p suprisin[st]

eδp(rminus s)

Eρ(r)p

H (106)

Complexity 13

I3t le cpE 1113946

t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859Q2

2

2dr1113888 1113889

p2

le cpLp

f2K

p22 1113946

t

se2(δminus α)(tminus r)

c2(t r)minus β2 ρ2minus 2( )ρ2( )e

minus α ρ2+2( )ρ2c2(tr)dr1113888 1113889

p2

times suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se2(δminus α)(tminus r)

c0(t minus r)1113858 1113859minus β2 ρ2minus 2( )ρ2( )dr1113888 1113889

p2

le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0r

minus β2 ρ2minus 2( )ρ2( )(p(pminus 2))dr1113888 1113889

(pminus 2)2

1113946tminus s

0e

minus p(αminus δ)rdr1113888 1113889

le cp

Lp

f2

α minus δsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

I4t le cpE 1113946

t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

p

Hv2prime

(dz)dr

le cpLpg2

suprisin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0e

minus 2(αminus δ)rdr1113888 1113889

p2

+ 1113946tminus s

0e

minus p(αminus δ)rdr⎡⎣ ⎤⎦le cp

Lpg2

(α minus δ)cpsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(107)

Hence we have

suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H

+ cp1Lp

(α minus δ)cp2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(108)

erefore for αgt 0 large enough we can find 0lt δp lt αsuch that

cp1L

α minus δp1113872 1113873cp2lt 1 (109)

is implies that

suprisin[st]

epδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H (110)

Let δp pδp thanks to Lemma 4 we have

E vx(t s y) minus v

x(t s minus h y)

p

H

le cpeminus δp(tminus s)

y minus vx(s s minus h y)

p

H

le cpeminus δp(tminus s) 1 +x

p

H +yp

H + eminus δph

yp

H1113872 1113873

(111)

Because Lp(ΩH) has completeness for any pgep lets⟶ minus infin in (111) there exists ηx(t) isin Lp(ΩH) such that(99) holds en if we let h⟶infin in (111) we obtain (100)Using the Holder inequality we can get (99) and (100) holdsfor any pltp

If we take y1 y2 isin H use the same arguments forvx(t s y1) minus vx(t s y2) slt t we have

E vx

t s y1( 1113857 minus vx

t s y2( 1113857

p

Hle cpe

minus δp(tminus s)y1 minus y2

p

H slt t

(112)

Let s⟶ minus infin this means that the limit ηx(t) does notdepend on the initial condition y isin H

Finally we prove that ηx(t) is a mild solution of (79)Due to the limit ηx(t) does not depend on the initialcondition we can let initial condition y 0 For any slt t

and hgt 0 we have

vx(t s minus h 0) Uα2(t s)v

x(s s minus h 0) + ψα2 v

x(middot s minus h 0) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s minus h 0)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s minus h 0)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s minus h 0) z( 1113857 1113957N2prime

(dr dz)

(113)

Let h⟶infin on both sides due to (99) we can get forany slt t have

ηx(t) Uα2(t s)ηx

(s) + ψα2 ηx s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x ηx

(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x ηx

(r)( 1113857dwQ2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x ηx(r) z( 1113857 1113957N2prime

(dr dz)

(114)

is means that ηx(t) is a mild solution of (79)For any t isin R and x isin H we denote that the law of the

random variable ηx(t) is μxt and we introduce the transition

evolution operator

Pxstφ(y) Eφ v

x(t s y)( 1113857 slt t y isin H (115)

14 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

I3t le cpE 1113946

t

se

c2(tr)A2e(δminus α)(tminus r)

eδ(rminus s)

F2 r x vx(r s y)( 1113857 minus F2 r x v

x(r s minus h y)( 11138571113858 1113859Q2

2

2dr1113888 1113889

p2

le cpLp

f2K

p22 1113946

t

se2(δminus α)(tminus r)

c2(t r)minus β2 ρ2minus 2( )ρ2( )e

minus α ρ2+2( )ρ2c2(tr)dr1113888 1113889

p2

times suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946t

se2(δminus α)(tminus r)

c0(t minus r)1113858 1113859minus β2 ρ2minus 2( )ρ2( )dr1113888 1113889

p2

le cpLp

f2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0r

minus β2 ρ2minus 2( )ρ2( )(p(pminus 2))dr1113888 1113889

(pminus 2)2

1113946tminus s

0e

minus p(αminus δ)rdr1113888 1113889

le cp

Lp

f2

α minus δsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

I4t le cpE 1113946

t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

2

Hv2prime

(dz)dr1113888 1113889

p2

+ cpE1113946t

s1113946Z

ec2(tr)A2e

(δminus α)(tminus r)eδ(rminus s)

G2 r x vx(r s y)( 1113857 minus G2 r x v

x(r s minus h y)( 11138571113858 1113859

p

Hv2prime

(dz)dr

le cpLpg2

suprisin[st]

eδp(rminus s)

Eρ(r)p

H middot 1113946tminus s

0e

minus 2(αminus δ)rdr1113888 1113889

p2

+ 1113946tminus s

0e

minus p(αminus δ)rdr⎡⎣ ⎤⎦le cp

Lpg2

(α minus δ)cpsup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(107)

Hence we have

suprisin[st]

eδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H

+ cp1Lp

(α minus δ)cp2sup

risin[st]

eδp(rminus s)

Eρ(r)p

H

(108)

erefore for αgt 0 large enough we can find 0lt δp lt αsuch that

cp1L

α minus δp1113872 1113873cp2lt 1 (109)

is implies that

suprisin[st]

epδp(rminus s)

Eρ(r)p

H le cp y minus vx(s s minus h y)

p

H (110)

Let δp pδp thanks to Lemma 4 we have

E vx(t s y) minus v

x(t s minus h y)

p

H

le cpeminus δp(tminus s)

y minus vx(s s minus h y)

p

H

le cpeminus δp(tminus s) 1 +x

p

H +yp

H + eminus δph

yp

H1113872 1113873

(111)

Because Lp(ΩH) has completeness for any pgep lets⟶ minus infin in (111) there exists ηx(t) isin Lp(ΩH) such that(99) holds en if we let h⟶infin in (111) we obtain (100)Using the Holder inequality we can get (99) and (100) holdsfor any pltp

If we take y1 y2 isin H use the same arguments forvx(t s y1) minus vx(t s y2) slt t we have

E vx

t s y1( 1113857 minus vx

t s y2( 1113857

p

Hle cpe

minus δp(tminus s)y1 minus y2

p

H slt t

(112)

Let s⟶ minus infin this means that the limit ηx(t) does notdepend on the initial condition y isin H

Finally we prove that ηx(t) is a mild solution of (79)Due to the limit ηx(t) does not depend on the initialcondition we can let initial condition y 0 For any slt t

and hgt 0 we have

vx(t s minus h 0) Uα2(t s)v

x(s s minus h 0) + ψα2 v

x(middot s minus h 0) s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x v

x(r s minus h 0)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x v

x(r s minus h 0)( 1113857dw

Q2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x vx(r s minus h 0) z( 1113857 1113957N2prime

(dr dz)

(113)

Let h⟶infin on both sides due to (99) we can get forany slt t have

ηx(t) Uα2(t s)ηx

(s) + ψα2 ηx s( 1113857(t)

+ 1113946t

sUα2(t r)B2 r x ηx

(r)( 1113857dr

+ 1113946t

sUα2(t r)F2 r x ηx

(r)( 1113857dwQ2(r)

+ 1113946t

s1113946Z

Uα2(t r)G2 r x ηx(r) z( 1113857 1113957N2prime

(dr dz)

(114)

is means that ηx(t) is a mild solution of (79)For any t isin R and x isin H we denote that the law of the

random variable ηx(t) is μxt and we introduce the transition

evolution operator

Pxstφ(y) Eφ v

x(t s y)( 1113857 slt t y isin H (115)

14 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

where φ isinBb(H)Due to (91) and (99) for any pge 1 we have

suptisinR

E ηx(t)

p

Hle cp 1 +x

p

H1113872 1113873 x isin H (116)

so that

suptisinR

1113946H

yp

Hμxt (dy)le cp 1 +x

p

H1113872 1113873 x isin H (117)

According to the above conclusion by using the samearguments as ([25] Proposition 53) we know that thefamily μx

t1113864 1113865tisinR defines an evolution system of probabilitymeasures on H for equation (76) is means that for anyt isin R μx

t is a probability measure on H and it holds that

1113946H

Pxstφ(y)μx

s (dy) 1113946Hφ(y)μx

t (dy) slt t (118)

for every φ isin Cb(H) Moreover we also have

Pxstφ(y) minus 1113946

Hφ(y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le ce

minus δ1(tminus s) 1 +xH( 1113857 (119)

Lemma 6 Under (A1)ndash(A4) the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (120)

is tight in P(D([0 T]H))

Proof According to the definition of vx(middot s 0) for any tgt s

and hgt 0 we have

vx(t + h s 0) minus v

x(t s 0) Uα2(t + h t) minus I1113872 1113873v

x(t s 0)

+ ψα2 vx(middot s 0) t( 1113857(t + h) + 1113946

t+h

tUα2(t + h r)

middot B2 r x vx(r s 0)( 1113857dr + 1113946

t+h

tUα2(t + h r)

middot F2 r x vx(r s 0)( 1113857dw

Q2(r) + 1113946t+h

t

middot 1113946Z

Uα2(t + h r)G2 r x vx(r s 0) z( 1113857 1113957N2prime

(dr dz)

(121)Due to the assumption (A4) we know that the mappings

B2 F2 andG2 are linearly growing For any tgt s 0lt hlt 1and x isin H analogous to the proof of Lemma 3 and using theestimate (91) we can get

E vx(t + h s 0) minus v

x(t s 0)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873

(122)

where κ(p) is a function of p and satisfies κ(p)gt 0

From Lemma 5 we know that the limit ηx(t) does notdepend on the initial condition and we can get

E ηx(t + h) minus ηx

(t)

p

HleE ηx

(t + h) minus vx(t + h s 0)

p

H

+ E vx(t + h s 0) minus v

x(t s 0)

p

H+ E v

x(t s 0) minus ηx

(t)

p

H

(123)

en thanks to (100) and (122) let s⟶infin we have

E ηx(t + h) minus ηx

(t)

p

Hle cp h

κ(p)+ h1113872 1113873 1 +x

p

H1113872 1113873 (124)

Moreover we have proved that the family μxt1113864 1113865tisinR defines

an evolution system of probability measures for (76) Byusing Chebyshevrsquos inequality and (116) it yields

limn⟶infin

μxt ηx(t) ηx(t)

Hge n1113872 1113873le lim

n⟶infin

E ηx(t)

2H

n2

le limn⟶infin

c21 +x2H

n2 0

(125)

In addition for any tgt s hgt 0 and fixed ϵgt 0 accordingto the (124) we have

limh⟶0

μxt ηx(t) ηx(t + h) minus ηx(t)

Hge ϵ1113872 1113873

le limh⟶0

E ηx(t + h) minus ηx(t)

2H

ϵ2

le limh⟶0

c2hκ(2) + h( 1113857 1 +x2H1113872 1113873

ϵ2 0

(126)

According to eorem 132 in [43] (125) and (126)imply that the family of measures

Λ ≔ μxt t isin R x isin H1113864 1113865 (127)

is tight in P(D([0 T]H))

In order to get the averaged equation wemust ensure theexistence of the averaged coefficient B1 So we need theevolution family of measures satisfying some nice propertiesWe give the following assumption

(A5)

(a) e functions c2 R⟶ (0infin) andl2 R times O⟶ Rd are periodic with the sameperiod

(b) e families of functions

B1R ≔ b1(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

B2R ≔ b2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

FR ≔ f2(middot ξ σ) ξ isin O σ isin BR2(R)1113864 1113865

GR ≔ g2(middot ξ σ z) ξ isin O σ isin BR2(R) z isin Z1113864 1113865

(128)

are uniformly almost periodic for any Rgt 0

Remark 4 Similar with the proof of Lemma 62 in [25] weget that under (A5) for any Rgt 0 the families of functions

B1(middot x y) (x y) isin BHtimesH(R)1113864 1113865

B2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

F2(middot x y) (x y) isin BHtimesH(R)1113864 1113865

G2(middot x y z) (x y z) isin BHtimesH(R) times Z1113864 1113865

(129)

Complexity 15

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

are uniformly almost periodicAs we know above A2(middot) is periodic Remark 4 holds

and the family of measures Λ is tight By proceeding as [45]we can prove that the mapping

t⟼μxt t isin R x isin H (130)

is almost periodic

5 The Averaged Equation

Lemma 7 Under (A1)ndash(A5) for any compact set K sub H thefamily of functions

t isin R⟼1113946H

B1(t x y)μxt (dy) x isin K1113882 1113883 (131)

is uniformly almost periodic

Proof AsK is a compact set inH so it is bounded and thereexist some Rgt 0 such thatK sub BH(R) at is for any x isin K

we have xleR

Now let us define

Φ(t x) 1113946H

B1(t x y)μxt (dy) (t x) isin R times H (132)

en for any t τ isin R n isin N and x isin K according toassumption (A3) and equation (117) we can get

|Φ(t + τ x) minus Φ(t x)|le 1113946yH le n

B1(t + τ x y)μxt+τ(dy) minus 1113946

yH le nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946yH gtn

B1(t + τ x y)μxt+τ(dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868+ 1113946

yH gt nB1(t + τ x y)μx

t (dy)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ 1113946H

B1(t + τ x y)μxt (dy) minus 1113946

HB1(t x y)μx

t (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le supxisinKyH le n

B1(t + τ x y)

H 1113946H

μxt+τ minus μx

t( 1113857(dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

+c(1 +x)

n+ 1113946

HB1(t + τ x y) minus B1(t x y)

Hμ

xt (dy)

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

(133)

Fixed some n such that c(1 + x)nle ϵ3 en for anyt isin R due to the mapping t⟼μx

t and the families offunctions B1(middot x y) are almost periodic we can find someτ isin R such that

|Φ(t + τ x) minus Φ(t x)|lt ϵ (134)

So the function Φ(middot x) is almost periodic for any x isin KBy preceding as in Lemma 5 we can get that under

(A1)ndash(A4) for any fixed x1 x2 isin K and pge 1 there existscp gt 0 such that

supsltt

E vx1(t s 0) minus v

x2(t s 0)

p

Hle cp x1 minus x2

p

H (135)

According to (99) and (100) let s⟶ 0 it yields

suptisinR

E ηx1(t) minus ηx2(t)

p

Hle cp x1 minus x2

p

H (136)

Hence thanks to (A3) and (136) we can conclude thatfor any x1 x2 isin K we have

Φ tx1( 1113857 minus Φ tx2( 1113857

H le E B1 tx1ηx1(t)( 1113857 minus B1 tx2η

x2(t)( 1113857

H

le c x1 minus x2

H+ E ηx1(t) minus ηx2(t)

2H

1113874 111387512

1113888 1113889

le c x1 minus x2

H

(137)

is means that the family of functions Φ(t middot) t isin R

is equicontinuous about x

In view of eorem 210 in [32] the above conclusionsindicate that Φ(middot x) x isin K is uniformly almost periodicHence the proof is complete

According to eorem 34 in [25] we define

B1(x) ≔ limT⟶infin

1T

1113946T

01113946H

B1(t x y)μxt (dy)dt x isin H

(138)

And thanks to (A3) and (117) we have that

B1(x)

Hle c 1 +xH( 1113857 (139)

Lemma 8 Under (A1)ndash(A5) for any Tgt 0 s σ isin R andx y isin H

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857dt minus B1(x)

H

lec

T1 +xH +yH( 1113857 + α(T x)

(140)

for some mapping α [0infin) times H⟶ [0infin) such that

16 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

supTgt0

α(T x)le c 1 +xH( 1113857 x isin H (141)

And for any compact set K sub H we have

limT⟶infin

supxisinK

α(T x) 0 (142)

Proof We denote

ψxB1(t y) ≔ B1(t x y) minus 1113946

HB1(t x w)μx

t (dw) (143)

So

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt1113888 1113889

2

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857ψx

B1 t vx(t s y)( 11138571113858 1113859dtdr

2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857P

xrtψ

xB1 r v

x(r s y)( 11138571113960 1113961dtdr

le2

T2 1113946s+T

s1113946

s+T

rE ψx

B1 r vx(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

1113872 111387312dtdr

(144)

Due to (A3) (91) and (117) we have

E ψxB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682 le cE B1 r x v

x(r s y)( 1113857

2H

+ cE 1113946H

B1(r x w)

Hμxr (dw)1113874 1113875

2

le c 1 +x2H + E v

x(r s y)

2H

1113874 1113875

le c 1 +x2H + e

minus 2δ(rminus s)y

2H1113872 1113873

(145)

and according to (100) we get

E Pxrtψ

xB1 r v

x(r s y)( 1113857

111386811138681113868111386811138681113868111386811138682

E Pxrt B1 r x v

x(r s y)( 1113857 minus 1113946

HB1(r x w)μx

r (dw)1113876 1113877

2

H

E B1 t x vx(t s y) minus B1 t x ηx

(t)( 1113857( 1113857

2H

le cE vx(t s y) minus ηx

(t)

2Hle ce

minus δ2(tminus s) 1 +x2H +y

2H1113872 1113873

(146)

Let δ δ22 it follows

E1T

1113946s+T

sB1 t x v

x(t s y)( 1113857 minus 1113946

HB1(t x w)μx

t (dw)1113876 1113877dt

H

lec

T1 +xH +yH( 1113857 1113946

s+T

s1113946

s+T

re

minus δ(tminus s)dtdr1113888 1113889

12

lec

T1 +xH +yH( 1113857

(147)

Since the family of functions (131) is uniformly almostperiodic according to eorem 34 in [25] we can get thatthe limit

limT⟶infin

1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt (148)

converges to B1(x) uniformly with respect to s isin R andx isin K (K is a compact set in H) erefore if we define

α(T x) 1T

1113946s+T

s1113946H

B1(t x w)μxt (dw)dt minus B1(x)

H

(149)

we get the conclusion (142) Moreover thanks to the as-sumption (A3) equations (117) and (139) we also can get

α(T x)le1T

1113946s+T

s1113946H

B1(t x w)

Hμxt (dw)dt

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ B1(x)

Hle c 1 +xH( 1113857

(150)

Hence the proof is complete

Complexity 17

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

Now we introduce the averaged equation

du(t) A1(t)u(t) + B1(u(t))1113858 1113859dt + F1(t u(t))dwQ1(t)

+ 1113946Z

G1(t u(t) z) 1113957N1(dt dz) u(0) x isin H

(151)

Due to assumption (A3) we can easily get that themapping B1 is Lipschitz continuous So for any x isin H Tgt 0and pge 1 equation (151) admits a unique mild solutionu isin Lp(Ω D([0 T]H))

6 Averaging Principles

In this section we prove that the slow motion uϵ convergesto the averaged motion u as ϵ⟶ 0

Theorem 1 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isin [0 θ)) and y isin H for any Tgt 0 and ηgt 0 we have

limϵ⟶0

P suptisin[0T]

uϵ(t) minus u(t)

Hgt η1113888 1113889 0 (152)

where u is the solution of the averaged (151)

Proof For any h isin D(A1)capLinfin(O) we have

languϵ(t) hrangH langx hrangH + 1113946t

0langA1(r)uϵ(r) hrangHdr

+ 1113946t

0langB1 uϵ(r)( 1113857 hrangHdr

+ lang1113946t

0F1 r uϵ(r)( 1113857dw

Q1(r) hrangH

+ lang1113946t

01113946Z

G1 r uϵ(r) z( 1113857 1113957N1(dr dz) hrangH

+ Rϵ(t)

(153)

where

Rϵ(t) ≔ 1113946t

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

(154)

According to the proof in Section 3 we know that thefamily L(uϵ)1113864 1113865ϵisin(01] is tight in P(D([0 T]H)) Hence inorder to prove eorem 1 it is sufficient to provelimϵ⟶0Esuptisin[0T]|Rϵ(t)| 0

For any ϵgt 0 and some deterministic constant δϵ gt 0 wedivide the interval [0 T] in subintervals of the size δϵ In eachtime interval [kδϵ (k + 1)δϵ] k 0 1 lfloorTδϵrfloor we definethe following auxiliary fast motion 1113954vϵ

d1113954vϵ(t) 1ϵ

A2(t) minus α( 11138571113954vϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858 1113859dt

+1ϵ

radic F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857dωQ2(t)

+ 1113946Z

G2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 1113957Nϵ2(dt dz)

(155)

According to the definition of 1113954vϵ we know thatan analogous estimate to (31) holds So for any pge 1 wehave

1113946T

0E 1113954vϵ(t)

p

Hdtle cpT 1 +x

p

H +yp

H1113872 1113873 (156)

Lemma 9 Under (A1)ndash(A5) fixing x isin D((minus A1)θ)

(θ isint[n0q hθ)) and y isin H there exists a constant κgt 0 suchthat if

δϵ ϵlnϵminus κ

(157)

we have

limϵ⟶0

suptisin[0T]

E 1113954vϵ(t) minus vϵ(t)

p

H 0 (158)

Proof Let ϵgt 0 be fixed For k 0 1 lfloorTδϵrfloorand t isin [kδϵ (k + 1)δϵ] let ρϵ(t) be the solution of theproblem

dρϵ(t) 1ϵ

A2(t) minus α( 1113857ρϵ(t)dt +1ϵ

radic Kϵ(t)dωQ2(t)

+ 1113946Z

Hϵ(t z) 1113957Nϵ2(dt dz) ρϵ kδϵ( 1113857 0

(159)

where

Kϵ(t) ≔ F2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus F2 t uϵ(t) vϵ(t)( 1113857

Hϵ(t z) ≔ G2 t uϵ kδϵ( 1113857 1113954vϵ(t) z( 1113857 minus G2 t uϵ(t) vϵ(t) z( 1113857

(160)

We get

ρϵ(t) ψαϵ2 ρϵ kδϵ( 1113857(t) + Γϵ(t) + Ψϵ(t) t isin kδϵ (k + 1)δϵ1113858 1113859

(161)

where

Γϵ(t) 1ϵ

radic 1113946t

kδϵUαϵ2(t r)Kϵ(r)dw

Q2(r)

Ψϵ(t) 1113946t

kδϵ1113946Z

Uαϵ2(t r)Hϵ(r z) 1113957Nϵ2(dr dz)

(162)

If we denote Λϵ(t) ≔ 1113954vϵ(t) minus vϵ(t) and ϑϵ(t) ≔Λϵ(t) minus ρϵ(t) we have

dϑϵ(t) 1ϵ

A2(t) minus α( 1113857ϑϵ(t) + B2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 11138571113858

minus B2 t uϵ(t) vϵ(t)( 11138571113859dt

(163)

Due to Lemma 3 for αgt 0 large enough using Youngrsquosinequality we have

18 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

1p

ddt

ϑϵ(t)

p

Hle1ϵ lang c2(t)A2 + L2(t) minus α( 1113857ϑϵ(t) ϑϵ(t)rangH ϑϵ(t)

pminus 2H

+1ϵlangB2 t uϵ kδϵ( 1113857 1113954vϵ(t)( 1113857 minus B2 t uϵ(t) vϵ(t)( 1113857 ϑϵ(t)rangH ϑϵ(t)

pminus 2H

lec

ϵϑϵ(t)

p

Hminusαϵϑϵ(t)

p

H+

c

ϵuϵ kδϵ( 1113857 minus uϵ(t)

H ϑϵ(t)

pminus 1H

+c

ϵ1113954vϵ(t) minus vϵ(t)

H ϑϵ(t)

pminus 1H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵuϵ kδϵ( 1113857 minus uϵ(t)

p

H+

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

le minusα2ϵ

ϑϵ(t)

p

H+

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)pϵ + δϵ1113872 1113873 +

cp

ϵ1113954vϵ(t) minus vϵ(t)

p

H

(164)

Using the Gronwall inequality we get

ϑϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵ1113954vϵ(r) minus vϵ(r)

p

Hdr

(165)

By proceeding as Lemma 63 in [13] we prove that

E Γϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(166)

For the other stochastic term Ψϵ(t) using Kunitarsquos firstinequality and the Holder inequality we get

E Ψϵ(t)

p

Hle cpE

1ϵ

1113946t

kδϵ1113946Z

eminus (αϵ)(tminus r)

ec2(tr)ϵ( )A2Hϵ(r z)

2

Hv2(dz)dr1113888 1113889

p2

+cp

ϵE1113946

t

kδϵ1113946Z

eminus αϵ(tminus r)

eA2c2(tr)ϵ

Hϵ(r z)

p

Hv2(dz)dr

lecp

ϵp2E 1113946

t

kδϵe

minus (2αϵ)(tminus r)uϵ kδϵ( 1113857 minus uϵ(r)

2H

+ 1113954vϵ(r) minus vϵ(r)

2H

1113874 1113875dr1113888 1113889

p2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

Hdr

lecp

ϵp21113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr 1113946

t

kδϵe

minus (2αϵ)(ppminus 2)(tminus r)dr1113888 1113889

(pminus 2)2

+cp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1113946

t

kδϵE uϵ kδϵ( 1113857 minus uϵ(r)

p

H+ E 1113954vϵ(r) minus vϵ(r)

p

H1113872 1113873dr

lecp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873 +

cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(167)

According to Lemma 24 in [25] and equations(161)ndash(167) we obtain

E 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵ1 +x

p

θ +yp

H1113872 1113873 δβ(θ)p+1ϵ + δ2ϵ1113872 1113873

+cp

ϵ1113946

t

kδϵE 1113954vϵ(r) minus vϵ(r)

p

Hdr

(168)From the Gronwall lemma this meansE 1113954vϵ(t) minus vϵ(t)

p

Hle

cp

ϵδβ(θ)p+1ϵ + δ2ϵ1113872 1113873e

cpϵ( 1113857δϵ 1 +xp

θ +yp

H1113872 1113873

(169)

For t isin [0 T] selecting δϵ ϵlnϵminus κ then if we take

κlt β(θ)p(β(θ)p + 1 + cp)and1(2 + cp) we have (158)

Lemma 10 Under the same assumptions as inGeorem 1 forany Tgt 0 we have

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868 0 (170)

Proof According to the definition of B1 we get that themapping B1 H⟶ H is Lipschitz continuous Using as-sumption (A3) and Lemmas 3 and 9 we have

Complexity 19

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

limϵ⟶0

E suptisin[0T]

Rϵ(t)1113868111386811138681113868

1113868111386811138681113868le limϵ⟶0

E1113946T

0langB1 r uϵ(r) vϵ(r)( 1113857 minus B1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 hrangH

11138681113868111386811138681113868111386811138681113868dr

+ limϵ⟶0

E suptisin[0T]

1113946t

0langB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ(r)( 1113857 hrangHdr

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le limϵ⟶0

cThH suprisin[0T]

E uϵ(r) minus uϵ lfloorrδϵrfloorδϵ( 1113857

H + suprisin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ lfloorrδϵrfloorδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

hH 1113944

lfloorTδϵrfloor

k01113946

(k+1)δϵ

kδϵE B1 uϵ kδϵ( 1113857( 1113857 minus B1 uϵ(r)( 1113857

Hdr

le limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)ϵ + δϵ1113872 1113873 + sup

risin[0T]

E 1113954vϵ(r) minus vϵ(r)

H1113890 1113891

+ limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

+ limϵ⟶0

cThH 1 +xθ +yH( 1113857 δβ(θ)+1ϵ + δ2ϵ1113872 1113873 lfloorTδϵrfloor + 1( 1113857

(171)

So we have to show that

limϵ⟶0

1113944

lfloorTδϵrfloor

k0E 1113946

(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 0

(172)

If we set ζϵ δϵϵ we have

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 E 1113946

δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113954vϵ kδϵ + r( 1113857( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

E 1113946δϵ

0langB1 kδϵ + r uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(rϵ)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

δϵE1ζϵ

1113946ζϵ

0langB1 kδϵ + ϵr uϵ kδϵ( 1113857 1113957v

uϵ kδϵ( )vϵ kδϵ( )(r)1113874 1113875 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

(173)

where 1113957vuϵ(kδϵ)vϵ(kδϵ)(r) is the solution of the fast motion (76)with the initial datum given by vϵ(kδϵ) and the frozen slowcomponent given by uϵ(kδϵ) In addition the noises in (76)are independent of uϵ(kδϵ) and vϵ(kδϵ) According to theproof of Lemma 8 we get

E 1113946(k+1)δϵ

kδϵlangB1 r uϵ kδϵ( 1113857 1113954vϵ(r)( 1113857 minus B1 uϵ kδϵ( 1113857( 1113857 hrangHdr

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

le δϵc

ζϵ1 + E uϵ kδϵ( 1113857

H + E vϵ kδϵ( 1113857

H1113872 1113873hH

+ δϵhHEα ζϵ uϵ kδϵ( 1113857( 1113857

(174)

In Section 3 we proved that the family

uϵ kδϵ( 1113857 ϵ gt 0 k 0 lfloorTδϵrfloor1113864 1113865 (175)is tight en by proceeding as the proof of equation (821)in [25] we also can get that for any ηgt 0 there exists acompact set Kη sub H such that

Eα ζϵ uϵ kδϵ( 1113857( 1113857le supxisinKη

α ζϵ x( 1113857 +η

radicc 1 +xH +yH( 1113857

(176)

Due to the arbitrariness of η and according to Lemma 8we can get (172) Furthermore (170) holds

rough the above proof eorem 1 is established

Data Availability

No data were used to support this study

20 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was partly supported by the National NaturalScience Foundation of China under Grant no 11772255 theFundamental Research Funds for the Central Universitiesthe Research Funds for Interdisciplinary Subject ofNorthwestern Polytechnical University the Shaanxi Projectfor Distinguished Young Scholars the Shaanxi ProvincialKey RampD Program (nos 2020KW-013 and 2019TD-010)and the Seed Foundation of Innovation and Creation forGraduate Students in Northwestern Polytechnical Univer-sity (ZZ2018027)

References

[1] N Bogolyubov and Y Mitropolskii Asymptotic Methods inthe Geory of Nonlinear Oscillations Gordon and BreachScience Publishers New York NY USA 1961

[2] V M Volosov ldquoAveraging in systems of ordinary differentialequationsrdquo Russian Mathematical Surveys vol 17 no 6pp 1ndash126 1962

[3] J G Besjes ldquoOn asymptotic methods for non-linear differ-ential equationsrdquo Journal de Mecanique vol 8 p 357 1969

[4] R Khasminskii ldquoOn the averaging principle for stochasticdifferential Ito equationsrdquo Kybernetika vol 4 pp 260ndash2791968

[5] D Givon ldquoStrong convergence rate for two-time-scale jump-diffusion stochastic differential systemsrdquoMultiscale Modelingamp Simulation vol 6 no 2 pp 577ndash594 2007

[6] M Freidlin and A Wentzell Random Perturbations of Dy-namical Systems Springer Science and BusinessMedia BerlinHeidelberg 2012

[7] J Duan andWWang Effective Dynamics of Stochastic PartialDifferential Equations Elsevier Berlin Heidelberg 2014

[8] W F ompson R A Kuske and A H Monahan ldquoSto-chastic averaging of dynamical systems with multiple timescales forced with α-Stable noiserdquo Multiscale Modeling ampSimulation vol 13 no 4 pp 1194ndash1223 2015

[9] Y Xu J Duan and W Xu ldquoAn averaging principle forstochastic dynamical systems with Levy noiserdquo Physica DNonlinear Phenomena vol 240 no 17 pp 1395ndash1401 2011

[10] Y Xu B Pei B Pei and R Guo ldquoStochastic averaging forslow-fast dynamical systems with fractional Brownian mo-tionrdquo Discrete amp Continuous Dynamical Systems-B vol 20no 7 pp 2257ndash2267 2015

[11] Y Xu B Pei and J-L Wu ldquoStochastic averaging principle fordifferential equations with non-Lipschitz coefficients drivenby fractional Brownian motionrdquo Stochastics and Dynamicsvol 17 no 02 p 1750013 2017

[12] S Cerrai ldquoA Khasminskii type averaging principle for sto-chastic reaction-diffusion equationsrdquo Ge Annals of AppliedProbability vol 19 no 3 pp 899ndash948 2009

[13] S Cerrai ldquoAveraging principle for systems of reaction-dif-fusion equations with polynomial nonlinearities perturbed bymultiplicative noiserdquo SIAM Journal on Mathematical Anal-ysis vol 43 no 6 pp 2482ndash2518 2011

[14] W Wang and A J Roberts ldquoAverage and deviation for slow-fast stochastic partial differential equationsrdquo Journal of Dif-ferential Equations vol 253 no 5 pp 1265ndash1286 2012

[15] B Pei Y Xu and G Yin ldquoAveraging principles for SPDEsdriven by fractional Brownian motions with random delaysmodulated by two-time-scale markov switching processesrdquoStochastics and Dynamics vol 18 no 4 p 1850023 2018

[16] B Pei Y Xu and J-L Wu ldquoStochastic averaging for sto-chastic differential equations driven by fractional Brownianmotion and standard Brownian motionrdquo Applied Mathe-matics Letters vol 100 p 106006 2020

[17] B Pei Y Xu Y Xu and Y Bai ldquoConvergence of p-th mean inan averaging principle for stochastic partial differentialequations driven by fractional Brownian motionrdquo Discrete ampContinuous Dynamical Systems-B vol 25 no 3 pp 1141ndash1158 2020

[18] Z Li and L Yan ldquoStochastic averaging for two-time-scalestochastic partial differential equations with fractionalbrownian motionrdquo Nonlinear Analysis Hybrid Systemsvol 31 pp 317ndash333 2019

[19] P Gao ldquoAveraging principle for the higher order nonlinearschrodinger equation with a random fast oscillationrdquo Journalof Statistical Physics vol 171 no 5 pp 897ndash926 2018

[20] P Gao ldquoAveraging principle for stochastic Kuramoto-Siva-shinsky equation with a fast oscillationrdquo Discrete amp Contin-uous Dynamical Systems-A vol 38 no 11 pp 5649ndash56842018

[21] P Gao ldquoAveraging principle for multiscale stochastic Klein-Gordon-Heat systemrdquo Journal of Nonlinear Science vol 29no 4 pp 1701ndash1759 2019

[22] V Chepyzhov and M Vishik Attractors for Equations ofMathematical Physics American Mathematical SocietyProvidence Rhode Island USA 2002

[23] A Carvalho J Langa and J Robinson Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems SpringerScience and Business Media Berlin Germany 2012

[24] J E Bunder and A J Roberts ldquoResolution of subgridmicroscale interactions enhances the discretisation ofnonautonomous partial differential equationsrdquo AppliedMathematics and Computation vol 304 pp 164ndash1792017

[25] S Cerrai and A Lunardi ldquoAveraging principle for nonau-tonomous slow-fast systems of stochastic reaction-diffusionequations the almost periodic caserdquo SIAM Journal onMathematical Analysis vol 49 no 4 pp 2843ndash2884 2017

[26] D Applebaum Levy Processes and Stochastic CalculusCambridge University Press Cambridge UK 2009

[27] J Duan An Introduction to Stochastic Dynamics CambridgeUniversity Press Cambridge UK 2015

[28] S Peszat and J Zabczyk Stochastic Partial DifferentialEquations with Levy Noise An Evolution Equation ApproachCambridge University Press Cambridge UK 2007

[29] J Bertoin Levy Processes Cambridge University PressCambridge UK 1998

[30] X Zhang Y Xu B Schmalfuszlig and B Pei ldquoRandomattractors for stochastic differential equations driven by two-sided Levy processesrdquo Stochastic Analysis and Applicationsvol 37 no 6 pp 1028ndash1041 2019

[31] Z Li L Yan and L Xu ldquoStepanov-like almost automorphicsolutions for stochastic differential equations with Levynoiserdquo Communications in Statistics - Geory and Methodsvol 47 no 6 pp 1350ndash1371 2018

[32] A Fink ldquoAlmost periodic differential equationsrdquo LectureNotes in Mathematics Vol 377 Springer-Verlag BerlinGermany 1974

[33] B Boufoussi and S Hajji ldquoSuccessive approximation ofneutral functional stochastic differential equations with

Complexity 21

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity

jumpsrdquo Statistics amp Probability Letters vol 80 no 5-6pp 324ndash332 2010

[34] J Luo ldquoFixed points and exponential stability of mild solu-tions of stochastic partial differential equations with delaysrdquoJournal of Mathematical Analysis and Applications vol 342no 2 pp 753ndash760 2008

[35] J Luo and T Taniguchi ldquoe existence and uniqueness fornon-Lipschitz stochastic neutral delay evolution equationsdriven by Poisson jumpsrdquo Stochastics and Dynamics vol 9no 1 pp 135ndash152 2009

[36] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer-Verlag BerlinGermany 1983

[37] G D Prato and J Zabczyk Stochastic Equations in InfiniteDimensions Cambridge University Press Cambridge UK2014

[38] L C Evans Partial Differential Equations American Math-ematical Society Cambridge UK 2nd edition 2010

[39] H Amann Linear and Quasilinear Parabolic Problems Bir-khaeuser Basel Switzerland 1995

[40] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer Science amp BusinessMedia Berlin Germany 2010

[41] S Cerrai ldquoStochastic reaction-diffusion systems with multi-plicative noise and non-Lipschitz reaction termrdquo ProbabilityGeory and Related Fields vol 125 no 2 pp 271ndash304 2003

[42] B Pei Y Xu and J-L Wu ldquoTwo-time-scales hyperbolic-parabolic equations driven by Poisson random measuresexistence uniqueness and averaging principlesrdquo Journal ofMathematical Analysis and Applications vol 447 no 1pp 243ndash268 2017

[43] P Billingsley Convergence of Probability Measures JohnWiley amp Son Berlin Germany 1968

[44] S Cerrai ldquoAsymptotic behavior of systems of stochasticpartial differential equations with multiplicative noise sto-chastic partial differential equations and applications-viirdquo2006

[45] G D Prato and C Tudor ldquoPeriodic and almost periodicsolutions for semilinear stochastic equationsrdquo StochasticAnalysis and Applications vol 13 pp 13ndash33 1995

22 Complexity