Averaging principle for systems of RDEs with polynomial

nonlinearities perturbed by multiplicative noise ∗

Sandra Cerrai†

Department of MathematicsUniversity of MarylandCollege Park, MD 20742

USA

August 21, 2011

Abstract

We prove the validity of an averaging principle for a class of systems of slow-fast reaction-diffusion equations with the reaction terms in both equations having polynomial growth,perturbed by a noise of multiplicative type. The models we have in mind are the stochasticFitzhugh-Nagumo equation arising in neurophysiology and the Ginzburg-Landau equationarising in statistical mechanics.

1 Introduction

In a series of recent papers ([6], written in collaboration with M. Freidlin, and [7] and [8]) wehave studied the validity of an averaging principle and the normal deviations of the slow motionfrom the averaged motion for the following class of systems of stochastic partial differentialequations of reaction-diffusion type on a bounded domain D of Rd, with d ≥ 1,

∂uε∂t

(t, ξ) = A1uε(t, ξ) + b1(ξ, uε(t, ξ), vε(t, ξ)) + g1(ξ, uε(t, ξ), vε(t, ξ))∂wQ1

∂t(t, ξ),

∂vε∂t

(t, ξ)=1

ε[(A2 − α)vε(t, ξ)+b2(ξ, uε(t, ξ), vε(t, ξ))]+

1√εg2(ξ, uε(t, ξ), vε(t, ξ))

∂wQ2

∂t(t, ξ),

uε(0, ξ) = x(ξ), vε(0, ξ) = y(ξ), ξ ∈ D,

N1uε (t, ξ) = N2vε (t, ξ) = 0, t ≥ 0, ξ ∈ ∂D,(1.1)

(here ε is a small positive parameter and α is a sufficiently large fixed constant). More precisely,we have proved that the slow motion uε is weakly convergent in C([0, T ];L2(D)), as ε ↓ 0,

∗Key words and phrases: Stochastic reaction diffusion equations, invariant measures, ergodic and stronglymixing processes, averaging principle.†Partially supported by the NSF grant DMS0907295 “Asymptotic Problems for SPDE’s”.

1

to the solution u of the so-called averaged equation, obtained by taking the average of thecoefficients b1 and g1 with respect to the invariant measure of the fast equation, with frozenslow component. Moreover, we have studied the normalized difference zε := (uε − u)/

√ε and

we have proved that it is weakly convergent in C([0, T ];L2(D)) to a process z, which is givenin terms of a Gaussian processes whose covariance is explicitly described.

In those articles, the stochastic perturbations are of multiplicative type and wQ1 and wQ2

are Gaussian noises, which are white in time and colored in space, in the case of space dimensiond > 1, with covariance operators Q1 and Q2. A1 and A2 are second order uniformly ellipticoperators endowed with the boundary conditions N1 and N2, respectively. The coefficientsb1, b2 and g1, g2 are suitable real-valued functions defined on D ×R2 which are assumed to beLipschitz continuous and in particular to have linear growth.

The Lipschitz-continuity assumption for the coefficients in system (1.1) has been crucialthroughout the quoted papers [6], [7] and [8] for several different reasons.

First of all the Lipschitz-continuity of coefficients implies that system (1.1) is well posedin the space L2(D) and hence we can work in the framework of square integrable functions,where, as a matter of fact, the use of stochastic analysis tools is simpler. In particular theproof of all a priori bounds for the slow and the fast motions uε and vε, which in any caserequires a lot of work, is less complicate.

Secondly, and most importantly, if the coefficients of the slow and of the fast motionequations are Lipschitz-continuous, then under the assumption that the deterministic part ofthe fast motion is asymptotically stable and the noisy perturbation is not too big, it is possibleto show that the averaged coefficients are Lipschitz-continuous and hence the averaged equationis well posed in L2(D).

Moreover, the Lipschitz continuity of coefficients makes the proof of the convergence of uεto u easier, as we can apply in several situations arguments based on the use of Gronwall’sLemma.

But on the other hand, it is clear that the Lipschitz-continuity assumption for the coeffi-cients in system (1.1) is too restrictive and does not allow to study the validity of an averagingprinciple in many relevant situations in which for example the coefficients have polynomialgrowth. One of this situations is represented by systems of reaction-diffusion equations ofFitzhugh-Nagumo or Ginzburg-Landau type, perturbed by a Gaussian noise. Such systemsarise in many fields of biological and physical sciences and have attracted considerable at-tention. One important application in biology is to the study of neurophysiology, with theHodgkin-Huxley system, whereas in physics similar equations arise in statistical mechanicswith the Ginzburg-Landau equation (see e.g. [26] for a mathematical introduction).

Our purpose in this paper is to cover such a gap and provide a way to study the averagingprinciple for a wide class of systems of two reaction-diffusion equations in a bounded domain Dof R d, for any d ≥ 1, perturbed by a multiplicative noise, in which the slow and the fast motionsare both described by an equation having polynomially growing and monotone reaction termsb1 and b2 (for all details see Hypotheses 2, 3 and (3)and Section 2).

A simple example of the reaction coefficients we have in mind is given by

b1(ξ, u, v) = f1(ξ, p1(ξ, u) + k1(ξ, u, v)),

andb2(ξ, u, v) = f2(ξ, p2(ξ, v) + k2(ξ, u)).

2

Here f1, f2 : D × R → R are continuous functions, and for any ξ ∈ D the functions f1(ξ, ·)and f2(ξ, ·) are of class C1, with non-negative and uniformly bounded derivatives. The func-tions p1(ξ, ·) and p2(ξ, ·) are polynomials of odd degree, having the leading coefficient strictlynegative, namely

pi(ξ, s) = −γi(ξ)s2ki+1 +

2ki∑j=1

γi,j(ξ)sj , i = 1, 2,

with

infξ∈ D

γi(ξ) =: γi > 0.

Finally, the mappings k1 : D × R2 → R and k2 : D × R → R are continuous functions, suchthat k1(ξ, ·) and k2(ξ, ·) are locally Lipschitz-continuous and have linear growth, uniformlywith respect to ξ ∈ D.

Moreover, in the present paper the diffusion coefficients g1 : D×R→ R and g2 : D×R2 → Rare continuous, with g1(ξ, ·) and g2(ξ, ·) Lipschitz-continuous, uniformly with respect to ξ ∈ D,and finally, α is a positive constant which has to be assumed large enough, depending on b2and g2.

The key object in the analysis of averaging for system (1.1) is the fast motion equationwith frozen slow component x ∈ C(D)

∂vx,y

∂t(t, ξ) = (A2 − α)vx,y(t, ξ) + b2(ξ, x(ξ), vx,y(t, ξ)) + g2(ξ, x(ξ), vx,y(t, ξ))

∂wQ2

∂t(t, ξ),

vx,y(0, ξ) = y(ξ), ξ ∈ D, N2vx,y(t, ξ) = 0, t ≥ 0, ξ ∈ ∂D.

(1.2)In Section 4 we study problem (1.2) in C(D) and we introduce the corresponding transitionsemigroup P xt , for any fixed x ∈ C(D). We prove that, as a consequence of our assumptions,P xt admits an invariant measure µx, having all moments finite.

Once we have the existence of an invariant measure µx, the crucial point is proving that µx

is the unique invariant measure, which assures the proper convergence of P xt ϕ to the mean of ϕwith respect to µx, as time t goes to infinity, for any ϕ belonging to a suitable class of functionsdefined on C(D), with values in R. Moreover, it is crucial to show that µx is nicely dependingon x. To this purpose, we assume that the solution vx,y of problem (1.2) satisfies Hypotheses 5and 6, and we provide the example of relevant situations in which these conditions are satisfied.In particular, we would like to stress that these results about the ergodic behavior of P xt arecompletely new, due to the lack of Lipschitz-continuity of the reaction coefficient b2 and thepresence of a noise of multiplicative type, and they can be of interest in their own, for theirapplications to other problems.

As we have a good knowledge of the fast motion equation, next step is introducing theaveraged equation. To this purpose we notice that, as the reaction coefficients do not havelinear growth, system (1.1) cannot be studied in L2(D). Actually, the natural space to studythe problem is C(D), the space of continuous functions on D. This means in particular thatthe averaged equation has to be studied in C(D) and its coefficients have to be defined inC(D). As b1 is non-Lipschitz, the most delicate step is to introduce the averaged reaction

3

coefficient. The obvious candidate is

B(x) =

∫C(D)

B1(x, z)µx(dz), x ∈ C(D), (1.3)

where B1(x, z)(ξ) = b1(ξ, x(ξ), z(ξ)), for any x, z ∈ C(D) and ξ ∈ D. Clearly B is notLipschitz-continuous, and for this reason we have to prove that it satisfies some monotonicityand dissipativity conditions which he inherits from b1 and which assure the well posedness ofthe averaged equation

∂u

∂t(t, ξ) = A1u(t, ξ) + B(u(t))(ξ) +G(u(t))(ξ))

∂wQ1

∂t(t, ξ),

u(0, ξ) = x(ξ), ξ ∈ D, N1u(t, ξ) = 0, t ≥ 0, ξ ∈ ∂D,

(1.4)

in the space C(D). To this purpose, it is important to stress that in this paper we show thatthe averaged coefficient B, defined in (1.3) as a non-local operator, is in fact a local operatorin C(D), unlike the same operator defined in L2(D) (see [6] and [7]). Actually, due to the niceergodic properties of (1.2), for any x ∈ C(D) we obtain B(x)(ξ) at any fixed ξ ∈ D, as thepointwise limit of suitable time averages.

Here, as in [6] and [7], our goal is proving that for any fixed T > 0 and η > 0 it holds

limε→0

P(|uε − u|C([0,T ];C(D)) > η

)= 0.

The first step consists in proving that the family L(uε)ε∈ (0,1] is tight in C([0, T ];C(D))and to this purpose we have to prove uniform bounds for uε, also in spaces of Holder continuousfunctions.

Once we have tightness, we have weak convergence in C([0, T ];C(D)) of a subsequenceuεkk∈N to some u. Thus, we have to prove that u is the solution of the averaged equationand, as we have uniqueness for such an equation, we can conclude that the whole family uεconverges to u, as ε ↓ 0, and, as g1 depends only on the slow motion, the convergence in factis in probability.

In order to characterize the weak limit of the subsequence uεkk∈N as the solution of theaveraged equation, we have to prove that for any h ∈ D(A1)

limε→0

E supt∈ [0,T ]

|Rhε (t)|C([0,T ];C(D)) = 0, (1.5)

where

Rhε (t) =

∫Duε(t, ξ)h(ξ) dξ −

∫Dx(ξ)h(ξ) dξ −

∫ t

0

∫Duε(s, ξ)A1h(ξ) dξ ds

−∫ t

0

∫DB(uε(s))(ξ)h(ξ) dξ ds−

∫ t

0

∫D

[G1(uε(s)h](ξ)dwQ2(s, ξ)

=

∫ t

0

∫D

(B1(uε(s), vε(s))(ξ)− B(uε(s))(ξ)

)h(ξ) dξ ds.

4

The proof of (1.5), as in [7], is based on the Khasminskii discretization in time and one of thecrucial step is proving that

limε→0

supt∈ [0,T ]

E|vε(t)− vε(t)|pC(D)= 0, (1.6)

where vε is the solution of the problemdv(t) =

1

ε[(A2 − α)v(t) +B2(uε(kδε), v(t))] dt+

1√εG2(uε(kδε), v(t)) dwQ2(t),

v(kδε) = vε(kδε).,

in each time interval [kδε, (k + 1)δε], for k = 0, . . . , [T/δε], and for a suitable δε (which has tobe determined), converging to zero, as ε ↓ 0.

Due to the lack of Lipschitz-continuity of coefficients, the proof of (1.6) seems to be notpossible, and hence we have to localize the coefficients. But, as we do not have uniform boundswith respect to ε > 0 for the fast motion vε in C([0, T ];C(D)), we cannot localize with respectto both variables u and v, but only with respect to u. This means that in the proof of (1.6)we have to use in a suitable way the dissipativity of b2 with respect to v.

Averaging principle both for deterministically and for randomly perturbed systems, havinga finite number of degrees of freedom, has been studied by many authors, under differentassumptions and with different methods. The first rigorous results are due to Bogoliubov (see[2]). Further developments were obtained by Volosov, Anosov and Neishtadt (see [22] and [28])and by Arnold et al. (see [1]). All these references are for the deterministic case. Concerningthe stochastic case, it is worth quoting the paper by Khasminskii [15], the works of Brin,Freidlin and Wentcell (see [11], [12], [13]), Veretennikov (see [27]) and Kifer (see for example[16], [17], [18], [19]).

The case of systems with an infinite number of degrees of freedom is more open. Apartfrom few papers dealing with averaging for infinite dimensional systems (to this purpose werefer to the papers [25] by Seidler-Vrkoc and [21] by Maslowskii-Seidler-Vrkoc, concerning withaveraging for Hilbert-space valued solutions of stochastic evolution equations depending on asmall parameter, and to the paper [20] by Kuksin and Piatnitski concerning with averaging fora randomly perturbed KdV equation), the behavior of solutions of infinite dimensional systemson time intervals of order ε−1 has been studied in few other papers than [6], [7] and [8].

Recently, Roberts and Wong in [24] have studied averaging for SPDEs with non Lipschitzcoefficients. In their paper, unlike in the present paper, they have considered the particularsituation of a system of two reaction diffusion equations in an interval (so that d = 1), bothperturbed by an additive noise and it seems that they cannot deal with the case of white noisein space. Moreover, the only coefficient which is not Lipschitz-continuous is the reaction termin the slow equation and, due to some Sobolev embedding constraints which arise from thefact that they study the problem in L2, its growth is dominated by a polynomial of degree 3.

2 Set up

Let D be a bounded domain of Rd, with d ≥ 1, having a regular boundary. Throughoutthe paper, we shall denote by H the separable Hilbert space L2(D), endowed with the scalar

5

product

〈x, y〉H =

∫Dx(ξ)y(ξ) dξ,

and with the corresponding norm | · |H . We shall endow the product space H × H with thescalar product

〈x, y〉H×H =

∫D〈x(ξ), y(ξ)〉R2 dξ = 〈x1, y1〉H + 〈x2, y2〉H

and the corresponding norm | · |H×H .Next, we shall denote by E the Banach space C(D), endowed with the sup-norm

|x|E = supξ∈ D|x(ξ)|,

and the duality 〈·, ·〉E . The product space E × E shall be endowed with the norm

|x|E×E =(|x1|2E + |x2|2E

) 12 ,

and the corresponding duality 〈·, ·〉E×E . Finally, for any θ ∈ (0, 1) we shall denote by Cθ(D)the subspace of θ-Holder continuous functions, endowed with the norm

|x|Cθ(D) = |x|E + [x]θ = |x|E + supξ,η∈ Dξ 6=η

|x(ξ)− x(η)||ξ − η|θ

.

For any p ∈ [1,∞], with p 6= 2, the norm in Lp(D) and Lp(D)×Lp(D) will be both denotedby | · |p. If δ > 0, we will denote by | · |δ,p the norm in W δ,p(D)

|x|δ,p := |x|p +

∫D

∫D

|x(ξ)− x(η)|p

|ξ − η|δp+ddξ dη. (2.1)

Now, we introduce some notations which we will use in what follows (for all details we refere.g. to [3, Appendix A] and [4, Section 2]). Let x ∈ E and let ξx ∈ D such that |x(ξx)| = |x|E .We denote by δx the element of E? defined for any y ∈ E by

〈δx, y〉E :=

x(ξx)y(ξx)

|x|E, if x 6= 0

〈δ, y〉E , if x = 0,

(2.2)

where δ is any element of E? of norm 1. Notice that

δx ∈ ∂ |x|E := h ∈ E? ; |h|E? = 1, 〈h, x〉E = |x|E

and for any differentiable mapping u : [0, T ]→ E

d

dt

−|u(t)|E ≤

⟨u′(t), δu(t)

⟩E. (2.3)

6

Analogously, if x ∈ E × E, we denote by δx the element of (E × E)? defined for anyy ∈ E × E by

〈δx, y〉E×E :=

x1(ξx1)y1(ξx1) + x2(ξx2)y2(ξx2)

|x|E×E, if x 6= 0

〈δ, y〉E×E , if x = 0,

(2.4)

where ξx1 , ξx2 ∈ D are such that |xi(ξxi)| = |xi|E , for i = 1, 2, and δ is any element of (E×E)?

of norm 1. As above, we have

δx ∈ ∂ |x|E×E :=h ∈ (E × E)? ; |h|(E×E)? = 1, 〈h, x〉E×E = |x|E

,

and (2.3) holds true, with E replaced by E × E.

Now, let X be any Banach space. We shall denote by Bb(X) the space of bounded Borelfunctions ϕ : X → R. Bb(X) is a Banach space, endowed with the sup-norm

‖ϕ‖0 := supx∈X|ϕ(x)|.

Cb(X) shall be the subspace of uniformly continuous mappings. Moreover, we shall denote byL(X) the space of bounded linear operators on X and, in the case X is a Hilbert space, weshall denote by L2(X) the subspace of Hilbert-Schmidt operators, endowed with the norm

‖Q‖2 =√

Tr [Q?Q].

The stochastic perturbations in the slow and in the fast motion equations (1.1) are givenrespectively by the Gaussian noises ∂wQ1/∂t(t, ξ) and ∂wQ2/∂t(t, ξ), for t ≥ 0 and ξ ∈ D,which are assumed to be white in time and colored in space, in the case of space dimensiond > 1. Formally, the cylindrical Wiener processes wQi(t, ξ) are defined as the infinite sums

wQi(t, ξ) =∞∑k=1

Qiek(ξ)βk(t), i = 1, 2,

where ekk∈N is a complete orthonormal basis in H, βk(t)k∈N is a sequence of mutu-ally independent standard Brownian motions defined on the same complete stochastic basis(Ω,F ,Ft,P) and Qi is a bounded linear operator on H.

The operators A1 and A2 are second order uniformly elliptic operators, having continuouscoefficients on D, and the boundary operators N1 and N2 can be either the identity operator(Dirichlet boundary condition) or a first order operator satisfying a uniform nontangentialitycondition.

The realizations A1 and A2 in E of the differential operators A1 and A2 , endowed withthe boundary conditions N1 and N2, generate two analytic semigroups etA1 and etA2 , t ≥ 0.As in [7], we assume that the operators A1, A2 and Q1, Q2 satisfy the following conditions.

7

Hypothesis 1. For i = 1, 2 there exist a complete orthonormal system ei,kk∈N of H, which iscontained in L∞(D), and two sequences of non-negative real numbers αi,kk∈N and λi,kk∈Nsuch that

Ai ei,k = −αi,k ei,k, Qiei,k = λi,kei,k, k ≥ 1,

and

κi :=∞∑k=1

λρii,k |ei,k|2∞ <∞, ζi :=

∞∑k=1

α−βii,k |ei,k|2∞ <∞,

for some constants ρi ∈ (2,+∞] and βi ∈ (0,+∞) such that

βi(ρi − 2)

ρi< 1. (2.5)

For comments and examples concerning these assumptions on the operators Ai and Qi, werefer to [7, Remark 2.1].

For any t, δ > 0 and p ≥ 1 the semigroups etAi map Lp(D) into W δ,p(D) with

|etAix|δ,p ≤ ci (t ∧ 1)−δ2 |x|p, x ∈ Lp(D). (2.6)

By using the Sobolev Embedding Theorem and the Riesz-Thorin Theorem, this implies thatthe semigroups etAi map Lp(D) into Lq(D), for any 1 ≤ p ≤ q ≤ ∞, and

|etAix|q ≤ ci (t ∧ 1)− d(q−p)

2pq |x|p, x ∈ Lp(D). (2.7)

Finally, as W δ,p(D) embeds into Cθ(D), for any θ < δ − d/p, we get

|etAix|Cθ(D) ≤ ci (t ∧ 1)−θ2 |x|E . (2.8)

As far as the reaction coefficient b1 : D × R2 → R in the slow equation is concerned, weassume the following condition.

Hypothesis 2. 1. The mapping b1 : D × R2 → R is continuous and there exists m1 ≥ 1such that

supξ∈ D|b1(ξ, σ)| ≤ c (1 + |σ1|m1 + |σ2|) , σ = (σ1, σ2) ∈ R2. (2.9)

2. There exist c > 0 and θ ≥ 0 such that

supξ∈ D|b1(ξ, σ)− b1(ξ, ρ)| ≤ c

(1 + |σ|θ + |ρ|θ

)|σ − ρ|, σ, ρ ∈ R2. (2.10)

3. There exists c > 0 such that for any σ, h ∈ R2

supξ∈ D

(b1(ξ, σ + h)− b1(ξ, σ))h1 ≤ c |h1| (1 + |σ|+ |h|) . (2.11)

8

Example 2.1. Let h : D×R→ R be a continuous function such that h(ξ, ·) : R→ R is locallyLipschitz-continuous, uniformly with respect to ξ ∈ D. Moreover, assume that

supξ∈ D|h(ξ, s)| ≤ c (1 + |s|m) , s ∈ R, (2.12)

and

h(ξ, s)− h(ξ, t) = ρ(ξ, s, t)(s− t), ξ ∈ D, s, t ∈ R, (2.13)

for some ρ : D × R2 → R such that

supξ∈ Ds,t∈R

λ(ξ, s, t) <∞.

Moreover, let k : D × R2 → R be a continuous function, such that k(ξ, ·) : R2 → R has lineargrowth, is locally Lipschitz-continuous, uniformly with respect to ξ ∈ D.

Now, we fix any continuous function f : D×R→ R such that f(ξ, ·) is of class C1, for anyξ ∈ D, and

0 ≤ ∂f

∂s(ξ, s) ≤ c, (ξ, s) ∈ D × R,

for some c > 0. Then, if we define

b(ξ, σ) = f(ξ, h(ξ, σ1) + k(ξ, σ1, σ2)),

it is not difficult to check that conditions 1 and 3 in Hypothesis 2 are all satisfied. Moreover, ifwe assume that h and k are differentiable and their derivatives have polynomial growth, thencondition 2 is satisfied.

Next, let γ and γi be continuous functions from D into R, for i = 1, . . . , 2k, with

infξ∈ D

γ(ξ) > 0.

Then, it is possible to check that the function

h(ξ, s) := −γ(ξ)s2k+1 +2k∑i=1

γi(ξ)si,

satisfies conditions (2.12) and (2.13).

2

For the reaction term b2 : D × R2 → R in the fast equation, we assume the followingconditions.

Hypothesis 3. 1. The mapping b2 : D × R2 → R is continuous and there exists m2 ≥ 1such that

supξ∈ D|b2(ξ, σ)| ≤ c (1 + |σ1|+ |σ2|m2) , σ = (σ1, σ2) ∈ R2. (2.14)

9

2. The mapping b2(ξ, ·) : R2 → R is locally Lipschitz continuous, uniformly with respect toξ ∈ D. Moreover, for any R > 0 there exists LR > 0 such that

|σ1| ≤ R, |ρ1| ≤ R =⇒ supξ∈ Dσ2∈R

|b2(ξ, σ1, σ2)− b2(ξ, ρ1, σ2)| ≤ LR |σ1 − ρ1|. (2.15)

3. There exists c > 0 such that for any σ, h ∈ R2

supξ∈ D

(b2(ξ, σ + h)− b2(ξ, σ))h2 ≤ c |h2| (1 + |σ|+ |h|) . (2.16)

4. For any σ1, σ2, ρ2 ∈ R we have

b2(ξ, σ1, σ2)− b2(ξ, σ1, ρ2) = −λ(ξ, σ1, σ2, ρ2)(σ2 − ρ2), (2.17)

for some continuous function λ : D × R3 → [0,+∞).

Example 2.2. Let h : D×R→ R be as in Example 2.1. Moreover, assume that the functionρ in (2.13) satisfies

supξ∈ Ds,t∈R

ρ(ξ, t, s) ≤ 0, (2.18)

and the mapping k satisfies

supξ∈ Ds∈R

(k(ξ, s, t1)− k(ξ, s, t2)) (t1 − t2) ≤ 0. (2.19)

Then conditions 1, 2 and 3 in Hypothesis 3 are all satisfied by b2(ξ, σ) = f2(ξ, h(ξ, σ2)+k(ξ, σ)).Notice that (2.18) holds for

h(ξ, s) = −γ(ξ)s2k+1 +

2k∑j=1

γj(ξ)sj − λs,

for λ large enough.

Concerning the diffusion coefficients g1 and g2 we assume they satisfy the following condi-tions.

Hypothesis 4. 1. The mappings g1 : D × R → R and g2 : D × R2 → R are continuousand the mappings g1(ξ, ·) : R → R and g2(ξ, ·) : D × R2 → R are Lipschitz-continuous,uniformly with respect to ξ ∈ D.

2. It holdssupξ∈ D|g1(ξ, σ1)| ≤ c

(1 + |σ1|

1m1

)(2.20)

andsupξ∈ Dσ1∈R

|g2(ξ, σ1, σ2)| ≤ c(

1 + |σ2|1m2

). (2.21)

10

In what follows, for any x, y ∈ E and for i = 1, 2 we shall set

Bi(x, y)(ξ) := bi(ξ, x(ξ), y(ξ)), ξ ∈ D

and

B := (B1, B2).

Due to Hypothesis 2, the mappings B1 and B2 are well defined and continuous from E×Einto E, so that B : E × E → E × E is well defined and continuous. As the mappings b1 andb2 have polynomial growth, B is not well defined in H ×H.

In view of (2.9) and (2.14), for any x, y ∈ E we have

|B1(x, y)|E ≤ c(1 + |x|m1

E + |y|E), |B2(x, y)|E ≤ c

(1 + |x|E + |y|m2

E

), (2.22)

so that, in particular,

|B(x, y)|E×E ≤ c(1 + |x|m1

E + |y|m2E

), x, y ∈ E. (2.23)

As a consequence of (2.11) and (2.16), it is immediate to check that for any x, y, h, k ∈ E

〈Bi(x+ h, y + k)−Bi(x, y), δh〉E ≤ c (1 + |h|E + |k|E + |x|E + |y|E) , i = 1, 2, (2.24)

so that for any (x, y), (h, k) ∈ E × E⟨B(x+ h, y + k)−B(x, y), δ(h,k)

⟩E×E ≤ c (1 + |(h, k)|E×E + |(x, y)|E×E) . (2.25)

Moreover, from (2.17) we have

〈B2(x, y + k)−B2(x, y), δk〉E ≤ 0. (2.26)

Finally, in view of (2.10) we have

|B1(x1, y1)−B1(x2, y2)|E ≤ c(

1 + |(x1, y1)|θE×E + |(x2, y2)|θE×E)

(|x1 − x2|E + |y1 − y2|E) .

(2.27)

Next, for any x, y, z ∈ E we define

[G1(x)z](ξ) = g1(ξ, x(ξ))z(ξ), [G2(x, y)z](ξ) := g2(ξ, x(ξ), y(ξ))z(ξ), ξ ∈ D.

Due to Hypothesis 4, we have that the mappings

x ∈ E 7→ G1(x) ∈ L(E)

and

(x, y) ∈ E × E 7→ G2(x, y) ∈ L(E)

are Lipschitz-continuous, for i = 1, 2, so that the same is true for the mapping G = (G1, G2)defined on E × E with values in L(E × E).

11

3 Solvability and a-priori bounds for the slow-fast system

With the notations introduced in Section 2, system (1.1) can be rewritten in the followingabstract form.

duε(t) = [A1uε(t) +B1(uε(t), vε(t))] dt+G1(uε(t)) dw

Q1(t),

dvε(t) =1

ε[(A2 − α)vε(t) +B2(uε(t), vε(t))] dt+

1√εG2(uε(t), vε(t)) dw

Q2(t),(3.1)

with initial conditions uε(0) = x ∈ E and vε(0) = y ∈ E. We are here concerned with thesolvability of the system above in Lp(Ω;C((0, T ];E×E)∩L∞(0, T ;E×E)) and with uniformbounds for its solution.

As proved in [4, Theorem 5.3], under Hypotheses 1 to 4 for any ε > 0 and x, y ∈ E thereexists a unique adapted mild solution to problem (3.1) in Lp(Ω;C((0, T ];E×E)∩L∞(0, T ;E×E)), with T > 0 and p ≥ 1. This means that there exist two adapted processes uε and vε inLp(Ω;C((0, T ];E) ∩ L∞(0, T ;E)) such that

uε(t) = etA1x+

∫ t

0e(t−s)A1B1(uε(s), vε(s)) ds+

∫ t

0e(t−s)A1G1(uε(s)) dw

Q1(s),

and

vε(t) = et(A2−α)

ε y +1

ε

∫ t

0e(t−s) (A2−α)

ε B2(uε(s), vε(s)) ds

+1√ε

∫ t

0e(t−s) (A2−α)

ε G2(uε(s), vε(s)) dwQ2(s),

and such processes are unique.

Lemma 3.1. Under Hypotheses 1 to 4, for any p ≥ 1 and T > 0 there exists a constantcp,T > 0 such that for any x, y ∈ E and ε ∈ (0, 1]

E supt∈ [0,T ]

|uε(t)|pE ≤ cp,T(1 + |x|pE + |y|pE

), (3.2)

and

E∫ T

0|vε(t)|pE dt ≤ cp,T

(1 + |x|pE + |y|pE

). (3.3)

Proof. Let ε ∈ (0, 1] be fixed. We denote

Γ1,ε(t) :=

∫ t

0e(t−s)A1G1(uε(s)) dw

Q1(s), t ∈ [0, T ],

and we set Λ1,ε(t) := uε(t)− Γ1,ε(t). We have

d

dtΛ1,ε(t) = A1Λ1,ε(t) +B1(Λ1,ε(t) + Γ1,ε(t), vε(t)), Λ1,ε(0) = x,

12

and then, according to (2.22) and (2.24)

d

dt

−|Λ1,ε(t)|E ≤

⟨A1Λ1,ε(t), δΛ1,ε(t)

⟩E

+⟨B1(Λ1,ε(t) + Γ1,ε(t), vε(t))−B1(Γ1,ε(t), vε(t)), δΛ1,ε(t)

⟩E

+⟨B1(Γ1,ε(t), vε(t)), δΛ1,ε(t)

⟩E

≤ c |Λ1,ε(t)|E + c (1 + |Γ1,ε(t)|E + |vε(t)|E) + c |B1(Γ1,ε(t), vε(t))|E

≤ c |Λ1,ε(t)|E + c(1 + |Γ1,ε(t)|m1

E + |vε(t)|E).

By comparison, this implies

|uε(t)|E ≤ |Λ1,ε(t)|E + |Γ1,ε(t)|E

≤ c(t) |x|E + c(t) sups∈ [0,t]

|Γ1,ε(s)|m1E + c(t)

∫ t

0(1 + |vε(s)|E) ds,

so that for any p ≥ 1 we obtain

E sups∈ [0,t]

|uε(s)|pE

≤ cp(t)(1 + |x|pE

)+ cp(t)E sup

s∈ [0,t]|Γ1,ε(s)|pm1

E + cp(t)

∫ t

0E |vε(r)|pE ds.

As we are assuming (2.20), by proceeding as in [4, proof of Theorem 4.2 and Remark 4.3] andas in [7, Lemma 4.1] it is possible to show that for any T and q ≥ 1

E supt∈ [0,T ]

|Γ1,ε(t)|qE ≤ cq(T )

(1 +

∫ T

0E |uε(t)|

qm1E dt

),

for some continuous increasing function cq(t) which is clearly independent of ε and vanishes att = 0. Hence, we have

E sups∈ [0,t]

|uε(s)|pE ≤ cp(t)(

1 + |x|pE +

∫ t

0E |vε(s)|pE ds

)+ cp(t)

∫ t

0E |uε(s)|pE ds. (3.4)

Thus, in order to conclude the proof of (3.2) we have to estimate∫ t

0E |vε(s)|pE ds.

As before, if we define

Γ2,ε(t) =1√ε

∫ t

0e(t−s) (A2−α)

ε G2(uε(s), vε(s)) dwQ2(s)

13

and set Λ2,ε(t) := vε(t)− Γ2,ε(t), it follows

d

dtΛ2,ε(t) =

1

ε[(A2 − α)Λ2,ε(t) +B2(uε(t),Λ2,ε(t) + Γ2,ε(t))] , Λ2,ε(0) = y.

Thanks to (2.22) and (2.26), for any p ≥ 1 we have

1

p

d

dt

−|Λ2,ε(t)|pE ≤

1

ε

⟨(A2 − α)Λ2,ε(t), δΛ2,ε(t)

⟩E|Λ2,ε(t)|p−1

E

+1

ε

⟨B2(uε(t),Λ2,ε(t) + Γ2,ε(t))−B2(uε(t),Γ2,ε(t)), δΛ2,ε(t)

⟩E|Λ2,ε(t)|p−1

E

+1

ε

⟨B2(uε(t),Γ2,ε(t)), δΛ2,ε(t)

⟩E|Λ2,ε(t)|p−1

E

≤ − α2ε|Λ2,ε(t)|pE +

cpε

(1 + |Γ2,ε(t)|m2p

E + |uε(t)|pE).

Integrating both sides in time, it follows

|Λ2,ε(t)|pE ≤ |y|pE −

cp(t)

ε

∫ t

0|Λ2,ε(s)|pE ds+

cp(t)

ε

(1 + sup

s∈ [0,t]|uε(s)|pE

)

+cpε

∫ t

0|Γ2,ε(s)|m2p

E ds.

(3.5)

Thus, by comparison this yields∫ t

0|Λ2,ε(s)|pE ds ≤ cp(t) |y|

pE + cp(t)

(1 + sup

s∈ [0,t]|uε(s)|pE

)

+cp

∫ t

0|Γ2,ε(s)|m2p

E ds,

so that, ∫ t

0E |vε(s)|pE ds ≤ cp

∫ t

0E |Λ2,ε(s)|pE ds+ cp

∫ t

0E|Γ2,ε(s)|pE ds

≤ cp(t) |y|pE + cp(t)

(1 + E sup

s∈ [0,t]|uε(s)|pE

)+ cp

∫ t

0E |Γ2,ε(s)|m2p

E ds.

By using a factorization argument and by proceeding as in [7, proof of Proposition 4.2],due to (2.5) and (2.21) it is possible to prove that for any k ≥ 1

supε∈ (0,1]

∫ t

0E|Γ2,ε(s)|kE ds ≤ ck(t)

∫ t

0

(1 + E |vε(s)|

km2E

)ds,

14

for some continuous increasing function ck(t) such that ck(0) = 0. Then, we have∫ t

0E |vε(s)|pE ds ≤ cp(t)

(|y|pE + 1

)+ cp(t)E sup

s∈ [0,t]|uε(s)|pE + cm2p(t)

∫ t

0E |vε(s)|pE ds.

As cm2p(0) = 0, we can fix t0 > 0 such that cm2p(t) ≤ 1/2, for any t ≤ t0, so that∫ t

0E |vε(s)|pE ds ≤ 2 cp(t)

(|y|pE + 1

)+ 2 cp(t)E sup

s∈ [0,t]|uε(s)|pE , t ≤ t0. (3.6)

If we plug the inequality above into (3.4), for any ε ∈ (0, 1] we easily get

E sups∈ [0,t]

|uε(s)|pE ≤ cp(t)(1 + |x|pE + |y|pE

)+ cp(t)E sup

s∈ [0,t]|uε(t)|pE , t ≤ t0.

Now, from the computations above, it is immediate to check that the function cp(t) is con-tinuous and vanishes at t = 0, so that we can fix 0 < t1 ≤ t0 such that cp(t1) ≤ 1/2. Thisimplies

E sups∈ [0,t1]

|uε(s)|pE ≤ 2 cp(t1)(1 + |x|pE + |y|pE

).

As the same argument can be repeated in the intervals [t1, 2t1], [2t1, 3t1] etc., we can concludethat (3.2) is true.

Finally, if we plug (3.2) into (3.6), we immediately get (3.3).

Next, we show that the family L(uε)ε∈ (0,1] is tight in C([0, T ];E). To this purpose, in

order to use the Ascoli-Arzela Theorem, we need uniform bound for uε in L∞(0, T ;Cθ(D)), forsome θ > 0, and uniform bounds for the increments of the mapping t ∈ [0, T ] 7→ uε(t) ∈ E.

Proposition 3.2. Under Hypotheses 1 to 4, there exists θ > 0 such that for any θ ∈ [0, θ),x ∈ Cθ(D), y ∈ E and T > 0

supε∈ (0,1]

E |uε|L∞(0,T ;Cθ(θ)) ≤ cT(

1 + |x|Cθ(D) + |y|E). (3.7)

Proof. By proceeding as in [4, Proposition 4.5 and Remark 4.6] we have that there exists θ1 > 0and p ≥ 1 such that for any T > 0 and p ≥ p

E |Γ1,ε|pL∞(0,T ;Cθ1 (D))≤ cp(T )

(1 + E |uε|pL∞(0,T ;E)

). (3.8)

Next, according to (2.8) and (2.22), if θ < 1 we have∣∣∣∣∫ t

0e(t−s)A1B1(uε(s), vε(s)) ds

∣∣∣∣Cθ(D)

≤ c∫ t

0(t− s)−

θ2(1 + |uε(s)|m1

E + |vε(s)|E)ds

≤ c(t)

(1 + sup

t∈ [0,T ]|uε(s)|m1

E

)+

(∫ t

0s−θ ds

) 12(∫ t

0|vε(s)|2E ds

) 12

,

15

and, according to (3.2) and (3.3), this implies that for any p ≥ 2

E supt∈ [0,T ]

∣∣∣∣∫ t

0e(t−s)A1B1(uε(s), vε(s)) ds

∣∣∣∣pCθ(D)

≤ cp(T )

(1 + E sup

t∈ [0,T ]|uε(s)|pm1

E

)+ cp(T )

∫ T

0E |vε(s)|pE ds ≤ cp(T )

(1 + |x|pm1

E + |y|pE).

(3.9)Moreover, if we assume that x ∈ Cθ(D) we have

|etA1x|Cθ(D) ≤ c |x|Cθ(D).

Thus, thanks to (3.8) and (3.9), for any θ ≤ θ1 ∧ 1 and ξ, η ∈ D we get

supε∈ (0,1]

E supt∈ [0,T ]

|uε(t, ξ)− uε(t, η)|p ≤ cp(T )(

1 + |x|pm1

Cθ(D)+ |y|pE

)|ξ − η|pθ,

so that, due to the characterization of the space W δ,p(D) given in (2.1), we have that for anyδ < θ

supε∈ (0,1]

E supt∈ [0,T ]

|uε(t)|δ,p ≤ cp(T )(

1 + |x|m1

Cθ(D)+ |y|E

).

As W δ,p(D) embeds into C θ(D), for p large enough, with θ = θ1 ∧ 1, we get (3.7).

Proposition 3.3. Assume Hypotheses 1 to 4. Then for any θ > 0 there exists γ(θ) > 0 suchthat for any T > 0, p ≥ 2, x ∈ Cθ(D), y ∈ E and t, s ∈ [0, T ] we have

supε∈ (0,1]

E |uε(t)− uε(s)|pE ≤ cp(T )(

1 + |x|pm1

Cθ(D)+ |y|pE

)|t− s|γ(θ)p. (3.10)

In view of estimates (3.2) and (3.3), the proof of the proposition above turns out to beanalogous to the proof of Proposition 4.4 in [7], and we do not repeat it.

Now, according to Proposition 3.2 and 3.3, thanks to the Ascoli-Arzela Theorem and theGarcia-Rademich-Rumsey Theorem we can conclude that the following result holds true (formore details see e.g. [7, Corollary 4.5].

Theorem 3.4. Under Hypotheses 1 to 4, for any x ∈ Cθ(D), with θ > 0, and y ∈ E, thefamily L(uε)ε∈ (0,1] is tight in C([0, T ];E).

4 The fast equation

For any frozen slow component x ∈ E and any initial datum y ∈ E, we introduce the problem

dv(t) = [(A2 − α)v(t) +B2(x, v(t))] dt+G2(x, v(t)) dwQ2(t), v(0) = y. (4.1)

According to Hypotheses 1, 3 and 4, equation (4.1) is well defined in E. That is, for anyx, y ∈ E there exists a unique mild solution vx,y ∈ Lp(Ω;C((0, T ];E) ∩ L∞(0, T ;E)), with

16

p ≥ 1 and T > 0 (for a proof see e.g. [4]). This allows us to introduce the transition semigroupP xt associated with equation (4.1), which is defined by

P xt ϕ(y) = Eϕ(vx,y(t)), t ≥ 0, y ∈ E,

for any ϕ ∈ Bb(E).For any λ > 0, equation (4.1) can be rewritten as

dv(t) = [(A2 − λ)v(t) +B2,λ(x, v(t))] dt+G2(x, v(t)) dwQ2(t), v(0) = y,

whereB2,λ(x, y) = B2(x, y) + (λ− α) y.

In what follows, for any x ∈ E and u ∈ Lp(Ω;Cb((0, T ];E)) we shall set

Γxλ(u)(t) =

∫ t

0e(t−s)(A2−λ)G2(x, u(s)) dwQ2(s), (4.2)

and

Γx(u)(t) =

∫ t

0e(t−s)A2G2(x, u(s)) dwQ2(s).

As proved in [5, Lemma3.1], there exists p > 1 such that for any t > 0, p ≥ p and 0 < δ < λand for any u, v ∈ Lp(Ω;Cb((0, T ];E))

sups∈ [0,t]

eδspE |Γxλ(u)(s)− Γxλ(v)(s)|pE ≤ cp,1Lpg2

(λ− δ)cp,2sups∈ [0,t]

eδsp E |u(s)− v(s)|pE , (4.3)

where Lg2 is the Lipschitz constant of g2 and cp,1, cp,2 are two suitable positive constants.Moreover, according to (2.21), we have

sups∈ [0,t]

eδspE |Γxλ(u)(s)|pE ≤ cp,1Mpg2

(λ− δ)cp,2sups∈ [0,t]

eδsp(

1 + E |u(s)|pm2E

), (4.4)

where

Mg2 = supξ∈ D, σ∈R2

|g2(ξ, σ)|

1 + |σ2|1m2

(see [5, Remark 3.2]).

Proposition 4.1. Assume Hypotheses 1, 3 and 4. Then, there exists δ > 0 such that for anyx, y ∈ E and p ≥ 1

E |vx,y(t)|pE ≤ cp(

1 + e−δpt |y|pE + |x|pE), t ≥ 0. (4.5)

Proof. If we set zλ(t) := vx,y(t)−Γxλ(t), where Γxλ(t) = Γxλ(vx,y)(t) is the stochastic convolutiondefined in (4.2), and λ > α, thanks to (2.26) we have

d

dt

−|zλ(t)|E ≤

⟨(A2 − λ)zλ(t), δzλ(t)

⟩E

+⟨B2,λ(x, zλ(t) + Γxλ(t))−B2,λ(x,Γxλ(t)), δzλ(t)

⟩E

+⟨B2,λ(x,Γxλ(t)), δzλ(t)

⟩E≤ −α |zλ(t)|E + c

(1 + |x|E + |Γxλ(t)|m2

E

)+ (λ− α) |Γxλ(t)|E

≤ −α |zλ(t)|E + c(1 + |x|E + |Γxλ(t)|m2

E

)+ (λ− α)

m2m2−1 ,

17

last estimate following from the Young inequality. By comparison we get

|zλ(t)|E ≤ e−αt|y|E + c(

1 + |x|E + (λ− α)m2m2−1

)+ c

∫ t

0e−α(t−s)|Γxλ(s)|m2

E ds,

so that for any p ≥ 1

|vx,y(t)|pE ≤ cp|Γxλ(t)|pE + cp e

−αpt|y|pE

+cp

(1 + |x|pE + (λ− α)

pm2m2−1

)+ cp

(∫ t

0e−α(t−s)|Γxλ(s)|m2

E ds

)p.

Next, we fix 0 < δ < α/2. Then, thanks to (4.4), we get

e2δptE |vx,y(t)|pE ≤ cpe2δpt E |Γxλ(t)|pE + cp |y|pE + cpe

2δpt(

1 + |x|pE + (λ− α)pm2m2−1

)+cp

(p− 1

(α− 2δ)p

)p−1 ∫ t

0e2δps E |Γxλ(s)|pm2

E ds

≤ cpMpg2

(λ− 2δ)cp,2sups∈ [0,t]

e2δsp

(1 + E |vx,y(s)|

pm2E

)

+cp |y|pE + cpe2δpt

(1 + |x|pE + (λ− α)

pm2m2−1

)+cp

(p− 1

(α− 2δ)p

)p−1 Mpg2

(λ− 2δ)cp,2

∫ t

0supr≤s

e2δpr(1 + E |vx,y(r)|pE

)ds.

(4.6)

Now, if take

λ1 =(2cpM

pg2

) 1cp,2 + 2δ,

we have

cpMpg2

(λ− 2δ)cp,2≤ 1

2, λ ≥ λ1.

Thus, if we take λ2 ≥ λ1 such that

cp

(p− 1

(α− 2δ)p

)p−1 Mpg2

(λ− 2δ)cp,2≤ δp, λ ≥ λ2

from (4.6) we have

sups≤t

e2δpsE |vx,y(s)|pE

≤ 2 cp |y|pE + cpe2δpt

(1 + |x|pE + (λ2 − α)

pm2m2−1

)+ δp

∫ t

0supr≤s

e2δpr E |vx,y(r)|pE ds.

18

Due to the Gronwall Lemma this yields

sups≤t

e2δpsE |vx,y(s)|pE ≤ 2 cp |y|pE + cpe2δpt

(1 + |x|pE + (λ2 − α)

pm2m2−1

)

+δp

∫ t

0eδp(t−s)

(2 cp |y|pE + cpe

2δps

(1 + |x|pE + λ

pm2m2−1

2

))ds, t ≥ 0,

and this immediately implies (4.5).

As proved in [4, Theorem 6.2], there exists some θ > 0 such that for any a > 0 and x, y ∈ E

supt≥a

E |vx,y(t)|Cθ(D) ≤ ca (1 + |x|E + |y|E). (4.7)

This implies that, if x, y ∈ E, then for any a > 0 the family L(vx,y(t))t≥a is tight inP(E,B(E)) and hence there exists an invariant measure µx for the semigroup P xt associatedwith equation (4.1) in E.

In view of (4.5), the invariant measure has all moments finite.

Proposition 4.2. For any ρ ≥ 1 there exists cρ > 0 such that∫E|z|ρE µ

x(dz) ≤ cρ(1 + |x|ρE

). (4.8)

Proof. Due to (4.5), for any t ≥ 0 we have∫E|z|ρE µ

x(dz) =

∫EP xt |z|

ρE µ

x(dz) =

∫EE |vx,z(t)|ρE µ

x(dz)

≤ cρ(1 + |x|ρE

)+ cρ e

−δρt∫E|z|ρE µ

x(dz).

Therefore, if we take t > 0 such that

cρ e−δρt <

1

2,

we get immediately (4.8).

In what follows, we shall assume that the solution of problem (4.1) satisfies the followingconditions.

Hypothesis 5. There exists a function β : [0,∞)2 → [0,∞) such that

supx∈E

E |vx,y1(t)− vx,y2(t)|2E ≤ β(t, |y1 − y2|E) t ≥ 0, (4.9)

for any y1, y2 ∈ E. Moreover, there exists some κ ≥ 0 such that

supt,s≥0

β(t, s)

sκ<∞

and

limt→∞

sups≥0

β(t, s)

sκ= 0. (4.10)

19

Conditions (4.9) and (4.10) have important consequences regarding the ergodic behaviorof system (4.1). Actually, as shown in the next lemma, (4.9) implies that µx is the uniqueinvariant measure for P xt and it is also strongly mixing.

Lemma 4.3. Let ϕ : E → R such that

|ϕ(y)− ϕ(z)| ≤ cϕ |y − z|E(

1 + |y|θE + |z|θE), y, z ∈ E,

for some θ ≥ 0. Then, there exists κθ ≥ 0 such that for any x, y ∈ E and t ≥ 0∣∣∣∣P xt ϕ(y)−∫Eϕ(z)µx(dz)

∣∣∣∣ ≤ cϕ (1 + |y|κθE + |x|κθE)β(t), (4.11)

for a function β ∈ L∞(0,∞) such that

limt→∞

β(t) = 0.

Proof. We have∣∣∣∣P xt ϕ(y)−∫Eϕ(z)µx(dz)

∣∣∣∣ ≤ ∫E|P xt ϕ(y)− P xt ϕ(z)| µx(dz)

≤∫EE |ϕ(vx,y(t))− ϕ(vx,z(t))|µx(dz)

≤ cϕ∫E

(E |vx,y(t)− vx,z(t)|2E

) 12

(1 + E |vx,y(t)|2θ + E |vx,z(t)|2θ

) 12µx(dz)

Then, according to (4.5), (4.8) and Hypothesis 5, we have∣∣∣∣P xt ϕ(y)−∫Eϕ(z)µx(dz)

∣∣∣∣ ≤ cϕ ∫E

√β(t, |y − z|E)

(1 + |x|θE + |y|θE + |z|θE

)µx(dz)

≤ cϕ sups≥0

√β(t, s)

sκ

∫E|y − z|

κ2E

(1 + |x|θE + |y|θE + |z|θE

)µx(dz)

≤ cϕ sups≥0

√β(t, s)

sκ

(1 + |x|θ∨

κ2

E + |y|θ∨κ2

E

).

Therefore, if we set

β(t) :=

√β(t, s)

sκ, κθ := θ ∨ κ

2,

we conclude the proof of the lemma.

Now, we describe a couple of situations in which condition (4.9) is satisfied.

20

Example 4.4. Let us consider the following reaction term for the fast equation

b2(ξ, σ1, σ2) = f(ξ, p(ξ, σ2) + h(ξ, σ1)), (4.12)

where

p(ξ, s) = −γ(ξ)s2k+1 +

2k∑j=1

γj(ξ)sj − λs, (4.13)

with λ > 0 and

infξ∈ D

γ(ξ) > 0,

and h : D ×R→ R is continuous with h(ξ, ·) locally Lipschitz-continuous, with linear growth,uniformly with respect to ξ ∈ D. Assume that f : D × R → R is continuous, and f(ξ, ·) ∈C1(R) with

0 ≤ ∂f

∂s(ξ, s) ≤ c, ξ ∈ D, s ∈ R. (4.14)

We want to prove that condition (4.9) is satisfied by the solution of problem (4.1), whenb2 is given by (4.12).

To this purpose, if we define ρ(t) := vx,y1(t) − vx,y2(t), then we have that ρ solves thefollowing linear equation

dρ(t) = [(A2 − α)ρ(t)− J(t)ρ(t)] dt+K(t)ρ(t)dwQ2(t), ρ(0) = y1 − y2, (4.15)

where for any (t, ξ) ∈ [0,+∞)× D

J(t, ξ) := H(t, ξ)

(p(ξ, vx,y1(t, ξ))− p(ξ, vx,y2(t, ξ)

vx,y2(t, ξ)− vx,y1(t, ξ)

)with

H(t, ξ) :=

∫ 1

0

∂f

∂s(ξ, θ p(ξ, vx,y1(t, ξ)) + (1− θ) p(ξ, vx,y2(t, ξ)) + h(ξ, x)) dθ

and

K(t, ξ) =g2(ξ, x(ξ), vx,y1(t, ξ))− g2(ξ, x(ξ), vx,y2(t, ξ))

vx,y1(t, ξ)− vx,y2(t, ξ).

Notice that, due to (4.14), for λ large enough we have

J(t, ξ) ≥ 0, (t, ξ) ∈ [0,∞)× D, P− a.s. (4.16)

and, due to the Lipschitz continuity of g2(ξ, ·) : R2 → R, uniform with respect to ξ ∈ D,

sup(t,ξ)∈ [0,∞)×D

|K(t, ξ)| ≤ supξ∈ Dh∈R

[g2(ξ, h, ·)]Lip <∞, P− a.s. (4.17)

In addition to problem (4.23), we introduce the following other linear problem

dρ′(t) = (A2 − α)ρ′(t) dt+K(t)ρ′(t)dwQ2(t), ρ′(0) = y1 − y2, (4.18)

21

By adapting the arguments used in [5] to equation (4.18), we have that if we assume

supξ∈ Dh∈R

[g2(ξ, h, ·)]Lip << α,

then there exists some δ > 0 such that for any p ≥ 1

E |ρ′(t)|pE ≤ cp e−δpt|y1 − y2|pE , t ≥ 0. (4.19)

Next, by a comparison argument, it is possible to prove that

y1 ≥ y2 =⇒ P− a.s. ρ′(t, ξ) ≥ 0, t ≥ 0, ξ ∈ D.

Moreover, we have that the same is true for ρ, that is

y1 ≥ y2 =⇒ P− a.s. ρ(t, ξ) ≥ 0, t ≥ 0, ξ ∈ D.

Therefore, since in view of the sign condition (4.16) we have

y1 ≥ y2 =⇒ P− a.s. ρ(t, ξ) ≤ ρ′(t, ξ), t ≥ 0, ξ ∈ D, (4.20)

from (4.19) we can conclude

E |vx,y1(t)− vx,y2(t)|pE = E |ρ(t)|pE ≤ cp e−δpt|y1 − y2|pE , t ≥ 0, y1 ≥ y2.

Finally, in the general case y1, y2 ∈ E we have

E |vx,y1(t)− vx,y2(t)|pE ≤ 2p E |vx,y1(t)− vx,y1∧y2(t)|pE + 2p E |vx,y1∧y2(t)− vx,y2(t)|pE

≤ 2p cpe−δpt (|y1 − y1 ∧ y2|pE + |y1 ∧ y2 − y2|pE

)≤ cp e−δpt|y1 − y2|pE ,

and this yields (4.9), with

β(t, s) = c2 e−2δts.

2

Example 4.5. Assume that the diffusion coefficient g2 in the fast equation does not dependon the fast motion, that is

g2(ξ, σ1, σ2) = g2(ξ, σ1), ξ ∈ D, (σ1, σ2) ∈ R2.

Then, if we define as in the previous example ρ(t) := vx,y1(t)− vx,y2(t), we have

d

dtρ(t) = A2ρ(t) +B2(x, vx,y1(t))−B2(x, vx,y2(t)), ρ(0) = y1 − y2.

This yieldsd

dt

−|ρ(t)|E ≤

⟨B2(x, vx,y1(t))−B2(x, vx,y2(t)), δρ(t)

⟩E. (4.21)

22

Now, if we assume

supξ∈ D

(b2(ξ, σ1, σ2 + h2)− b2(ξ, σ1, σ2))h2 ≤ −a |h2|m2+1,

we have〈B2(x, y + k)−B2(x, y), δk〉E ≤ −a |k|

m2E ,

and then from (4.21) we getd

dt

−|ρ(t)|E ≤ −a |ρ(t)|m2

E .

If m2 > 1, by comparison this implies

|vx,y1(t)− vx,y2(t)|E ≤ |y1 − y2|E(

1 + a(m2 − 1)|y1 − y2|m2−1E t

)− 1m2−1

,

so that (4.9) follows for

β(t, s) = s2(1 + a(m2 − 1)sm2−1 t

)− 2m2−1 .

Notice that in this caselimt→∞

sups≥0

β(t, s) = 0.

In the case m2 = 1, instead of polynomial we have exponential decay to zero, as in theprevious example and so we get (4.9) with

β(t, s) = c2e−2αt s.

2

In order to prove the averaging result of this paper we have also to require that the solutionof (4.1) satisfies the following condition (n what follows, for any R > 0 we denote by BE(R)the ball in E of radius R).

Hypothesis 6. For any R > 0 there exists KR > 0 such that

x1, x2 ∈ BE(R) =⇒ lim supT→∞

1

T

∫ T

0E |vx1,0(t)− vx2,0(t)|2E dt ≤ KR |x1 − x2|2E . (4.22)

As for Hypothesis 5, we describe relevant situations in which the condition described inHypothesis 6 is satisfied.

Example 4.6. We consider the same situation described in Example 4.4, that is

b2(ξ, σ1, σ2) = f(ξ, p(ξ, σ2) + h(ξ, σ1)),

where p is the polynomial introduced in (4.13) and h : D × R → R is a continuous functionwhich is locally Lipschitz-continuous and has linear growth in the variable σ1, uniformly withrespect to ξ ∈ D.

23

In this case, if we define ρ(t) := vx1,y(t)− vx2,y(t) we have that ρ solves the following linearequation

dρ(t) = [(A2 − α)ρ(t)− J(t)ρ(t) + I] dt+K(t) dwQ2 , ρ(0) = 0, (4.23)

where

I(ξ) = H(t, ξ) (h(ξ, x1(ξ))− h(ξ, x2(ξ))) ,

and

J(t, ξ) := H(t, ξ)

(p(ξ, vx1,y(t, ξ))− p(ξ, vx2,y(t, ξ)

vx2,y(t, ξ)− vx1,y(t, ξ)

),

with H(t, ξ) defined by∫ 1

0

∂f

∂s(ξ, θ [p(ξ, vx1,y(t, ξ)) + h(ξ, x1(ξ))] + (1− θ) [p(ξ, vx2,y(t, ξ)) + h(ξ, x2(ξ))]) dθ,

and

K(t, ξ) = g2(ξ, x1(ξ), vx1,y(t, ξ))− g2(ξ, x2(ξ), vx2,y(t, ξ)).

As h(ξ, ·) is locally Lipschitz continuous, uniformly with respect to ξ ∈ D, for any R > 0there exists cR > 0 such that

x1, x2 ∈ BE(R) =⇒ |I|E ≤ cR |x1 − x2|E . (4.24)

Now, we denote by Γ(t) the solution of the problem

dΓ(t) = (A2 − α)Γ(t) dt+K(t) dwQ2(t), Γ(0) = 0.

By proceeding as in [4, Theorem 4.2 and Proposition 4.5], due to the Lipschitz continuiuty ofg2 it is possible to prove that for any t > 0 and p ≥ 1

E sups∈ [0,t]

|Γ(s)|pE ≤ cα,p(t) supξ∈ D

[g2(ξ, ·)]pLip

(|x1 − x2|pE + E sup

s∈ [0,t]|ρ(s)|pE

), (4.25)

for some function cα,p : [0,∞)→ [0,∞) such that

limα→∞

supt≥0

cα,p(t) = 0. (4.26)

Next, if we define z(t) := ρ(t)− Γ(t), we have that z solves the problem

dz(t) = (A2 − α)z(t)− J(t)z(t) + I − J(t)Γ(t), z(0) = 0.

For any t ≥ 0 we denote by ξt the point in D such that

|z(t, ξt)| = |z(t)|E

and ⟨h, δz(t)

⟩E

=1

|z(t)|Eh(ξt)z(t, ξt),

24

for any h ∈ E. Thus, as in view of (4.16) we have that for λ large enough J(t, ξt) ≥ 0, P-a.s.,we get

d

dt

−|z(t)|E ≤

⟨(A2 − α)z(t), δz(t)

⟩E−⟨J(t)z(t), δz(t)

⟩E

+⟨I, δz(t)

⟩E−⟨J(t)Γ(t), δz(t)

⟩E

≤ −(α+ J(t, ξt))|z(t)|E + |I|E + J(t, ξt)|Γ(t)|E .

By comparison this yields

|z(t)|E ≤1

α|I|E +

∫ t

0exp

(−∫ t

s(α+ J(r, ξr)) dr

)J(s, ξs)|Γ(s)|E ds,

and then

|z(t)|E ≤1

α|I|E + sup

s≤t|Γ(s)|E

∫ t

0exp

(−∫ t

sJ(r, ξr) dr

)J(s, ξs) ds

=1

α|I|E + sup

s≤t|Γ(s)|E

(1− exp

(−∫ t

0J(r, ξr) dr

))≤ 1

α|I|E + sup

s≤t|Γ(s)|E .

This yields,

sups≤t|ρ(t)|2E ≤

2

α2|I|2E + 8 sup

s≤t|Γ(s)|2E ,

so that, thanks to (4.24) and (4.25), we have that for any R > 0 and x1, x2 ∈ BE(R)

E sups≤t|ρ(t)|2E ≤

(2 c2

R

α2+ cα,2(t)

)|x1 − x2|2E + 8 cα,2(t) sup

ξ∈ D[g2(ξ, ·)]2Lip E sup

s∈ [0,t]|ρ(s)|2E

≤(

2c2R

α2+ |cα,2|2∞

)|x1 − x2|2E + 8 |cα,2|2∞ sup

ξ∈ D[g2(ξ, ·)]2Lip E sup

s∈ [0,t]|ρ(s)|2E .

Therefore, due to (4.26) we can fix α large enough such that

8 |cα,2|2∞ supξ∈ D

[g2(ξ, ·)]2Lip ≤1

2,

so that

E sups≤t|ρ(t)|2E ≤ 2

(2c2R

α2+ |cλ,2|2∞

)|x1 − x2|2E =: KR |x1 − x2|2E .

This clearly implies the validity of Hypothesis 6.

2

Example 4.7. If we consider the case covered in Example 4.5, then with arguments similar tothose used throughout Example 4.5 itself, it is possible to prove that Hypothesis 6 is satisfied.

2

25

5 The averaged equation

In this section we introduce the averaged equation. The key point is constructing the coeffi-cients and describing some of its properties.

For any fixed x ∈ E, we introduce the mapping

y ∈ E 7→ B1(x, y) ∈ E.

Due to our assumptions, the mapping B1(x, ·) : E → E is continuous and

|B1(x, y)|E ≤ c(1 + |x|m1

E + |y|E). (5.1)

Now, we define

B(x) :=

∫EB1(x, y)µx(dy), x ∈ E.

Due to the estimates above and to (4.8), the mapping B : E → E is well defined and

|B(x)|E ≤ c(1 + |x|m1

E

). (5.2)

Actually, in view of (5.1) we have

|B(x)|E ≤∫E|B1(x, y)|E µx(dy) ≤ c

(1 + |x|m1

E

)+ c

∫E|y|E µx(dy),

and then, thanks to (4.8), we have

|B(x)|E ≤ c(1 + |x|m1

E

)+ c1 (1 + |x|E) ,

which implies (5.2).As a consequence of (4.11), we can give the following characterization of B.

Lemma 5.1. Assume Hypotheses 1 to 5. Then, there exist some constants κ1, κ2 ≥ 0 suchthat for any T > 0, t ≥ 0 and x, y ∈ E and for any Λ ∈ E?

E∣∣∣∣ 1

T

∫ t+T

t〈B1(x, vx,y(s)),Λ〉E ds−

⟨B(x),Λ

⟩E

∣∣∣∣ ≤ α(T )(1 + |x|κ1E + |y|κ2E

)|Λ|E? , (5.3)

for some mapping α : [0,∞)→ [0,∞) such that

limT→∞

α(T ) = 0.

Proof. For any fixed Λ ∈ E? and x ∈ E, we denote by ΠxΛB1 the mapping

z ∈ E 7→ ΠxΛB1(z) :=

⟨B1(x, z)− B(x),Λ

⟩E∈ R.

By proceeding as in the proof of [7, Lemma 2.3], due to Markovianity of vx,y(t) we have

E(

1

T

∫ t+T

t〈B1(x, vx,y(s)),Λ〉E ds−

⟨B(x),Λ

⟩E

)2

=2

T 2

∫ t+T

t

∫ t+T

rE[Πx

ΛB1(vx,y(r))P xs−rΠxΛB1(vx,y(r))

]ds dr

≤ 2

T 2

∫ t+T

t

∫ t+T

r

(E |Πx

ΛB1(vx,y(r))|2) 1

2(E |P xs−rΠx

ΛB1(vx,y(r))|2) 1

2 ds dr.

(5.4)

26

Due to (2.22), (4.5) and (5.2), we have

E |ΠxΛB1(vx,y(r))|2 ≤ c

(1 + |x|2m1

E + E |vx,y(r)|2E)|Λ|2E?

≤ c(

1 + |x|2m1E + e−2δr|y|2E

)|Λ|2E? .

(5.5)

Moreover, as due to (2.27) we have

|〈B1(x, y),Λ〉E − 〈B1(x, z),Λ〉E | ≤ c |y − z|E(

1 + |x|θE + |y|θE + |z|θE)|Λ|E? ,

thanks to (4.11) we have

E |P xs−rΠxΛB1(vx,y(r))|2 ≤ c

(1 + |x|2(θ∨κθ)

E + |y|2(θ∨κθ)E

)|Λ|2E?β2(s− r).

So, if we plug the estimate above and estimate (5.5) into (5.4), we get

E(

1

T

∫ t+T

t〈B1(x, vx,y(s)),Λ〉E ds−

⟨B(x),Λ

⟩E

)2

≤ c(1 + |x|m1

E + |y|E) (

1 + |x|θ∨κθE + |y|θ∨κθE

)|Λ|2E?

1

T 2

∫ t+T

t

∫ t+T

rβ(s− r) ds dr,

so that, if we define

α(T ) =

(supt≥0

1

T 2

∫ t+T

t

∫ t+T

rβ(s− r)ds dr

) 12

,

we have

E∣∣∣∣ 1

T

∫ t+T

t〈B1(x, vx,y(s)),Λ〉E ds−

⟨B(x),Λ

⟩E

∣∣∣∣≤ c

(1 + |x|κ1E + |y|κ2E

)|Λ|E?α(T ),

for some constants κ1, κ2 ≥ 0. Therefore, in order to conclude the proof of the lemma, we haveto show that limT→∞ α(T ) = 0.

For any ε > 0 we can fix Tε > 0 such that β(s) ≤ ε, for s ≥ Tε. Thus, for any T > Tε wehave

1

T 2

∫ t+T

t

∫ t+T

rβ(s− r)ds dr =

1

T 2

∫ t+T

t

∫ t+T−r

0β(s)ds dr

=1

T 2

∫ t+T−Tε

t

∫ t+T−r

0β(s)ds dr +

1

T 2

∫ t+T

t+T−Tε

∫ t+T−r

0β(s)ds dr

=: I1,ε(t, T ) + I2,ε(t, T ).

27

For the first term I1,ε we have

I1,ε(t, T ) =1

T 2

∫ t+T−Tε

t

[∫ Tε

0β(s)ds+

∫ t+T−r

Tε

β(s) ds

]dr

≤ (T − Tε)TεT 2

|β|∞ +ε

T 2

∫ t+T−Tε

t(t+ T − Tε − r) dr =

(T − Tε)TεT 2

|β|∞ +ε(T − Tε)2

2T 2.

(5.6)For the second term I2,ε we have

I2,ε(t, T ) ≤ 1

T 2

∫ t+T

t+T−Tε|β|∞Tε dr =

|β|∞T 2ε

T 2,

and then, combining together this estimate with (5.6), we obtain

1

T 2

∫ t+T

t

∫ t+T

rβ(s− r)ds dr ≤ Tε

T|β|∞ +

ε

2+|β|∞T 2

ε

T 2, T > 0, t ≥ 0,

so that, due to the arbitrariness of ε, we conclude that

limT→∞

α(T ) = 0.

In view of the previous lemma, we have that for any x ∈ E and T > 0∣∣∣∣⟨ 1

T

∫ T

0EB1(x, vx,0(s)) ds− B(x),Λ

⟩E

∣∣∣∣≤ E

∣∣∣∣ 1

T

∫ T

0

⟨B1(x, vx,0(s)),Λ

⟩Eds−

⟨B(x),Λ

⟩E

∣∣∣∣ ≤ α(T ) (1 + |x|κ1) |Λ|E? .

This implies

limT→∞

∣∣∣∣ 1

T

∫ T

0EB1(x, vx,0(s)) ds− B(x)

∣∣∣∣E

= 0, (5.7)

and in particular we get

B(x)(ξ) = limT→∞

1

T

∫ T

0E b1(ξ, x(ξ), vx,0(t, ξ)) dt, ξ ∈ D.

Moreover, we can prove that B fulfills the following properties.

Lemma 5.2. Under Hypotheses 1 to 6, we have that B : E → E is locally Lipschitz-continuous.Moreover, for any x, h ∈ E⟨

B(x+ h)− B(x), δh⟩E≤ c (1 + |h|E + |x|E) . (5.8)

28

Proof. According to Lemma 5.1 and to (5.7), for any x1, x2 ∈ E we have

B(x1)− B(x2) = limT→∞

1

T

∫ T

0E(B1(x1, v

x1,0(s))−B1(x2, vx2,0(s))

)ds, in E. (5.9)

Now, by using (2.27) we have∣∣B1(x1, vx1,0(s))−B1(x2, v

x2,0(s))∣∣E

≤ c(

1 + |x1|θE + |x2|θE + |vx1,0(s)|θE + |vx2,0(s)|θE) (|x1 − x2|E + |vx1,0(s)− vx2,0(s)|E

),

and then, due to (4.5)∣∣E (B1(x1, vx1,0(s))−B1(x2, v

x2,0(s)))∣∣E

≤ c(

1 + |x1|θE + |x2|θE)(|x1 − x2|E +

(E|vx1,0(s)− vx2,0(s)|2E

) 12

).

Thanks to (4.22), this implies that for any T > 0 and R > 0, if x1, x2 ∈ BE(R) then∣∣∣∣ 1

T

∫ T

0E(B1(x1, v

x1,0(s))−B1(x2, vx2,0(s))

)ds

∣∣∣∣E

≤ c(

1 + |x1|θE + |x2|θE)(|x1 − x2|E +

1

T

∫ T

0

(E|vx1,0(s)− vx2,0(s)|2E

) 12 ds

)

≤ cKR

(1 + |x1|θE + |x2|θE

)|x1 − x2|E ≤ CR |x1 − x2|E ,

so that, due to (5.9) the local Lipschitz-continuity of B follows.Concerning (5.8), we have⟨B(x+ h)− B(x), δh

⟩E

= limT→∞

1

T

∫ T

0E⟨B1(x+ h, vx+h,0(s))−B1(x, vx,0(s)), δh

⟩Eds.

Now, due to (2.24) we have⟨B1(x+ h, vx+h,0(s))−B1(x, vx,0(s)), δh

⟩E

≤ c(

1 + |x|E + |h|E + |vx+h,0(s)|E + |vx,0(s)|E),

and then, thanks to (4.5),⟨B(x+ h)− B(x), δh

⟩E

≤ lim supT→∞

1

T

∫ T

0c(

1 + |x|E + |h|E + E|vx+h,0(s)|E + E|vx,0(s)|E)ds

≤ c (1 + |x|E + |h|E) .

29

Now, we can introduce the averaged equation

du(t) =[A1u(t) + B(u(t))

]dt+G(u(t)) dw(t), u(0) = x ∈ E. (5.10)

In view of Lemma 5.2 and of [4, Theorem 5.3], for any x ∈ E, T > 0 and p ≥ 1 equation (5.10)admits a unique mild solution u ∈ Lp(Ω;C((0, T ];E) ∩ L∞(0, T ;E)). In the next section wewill show that the slow motion uε converges in probability to the averaged motion u.

6 The averaging limit

In this last section we prove the following averaging result.

Theorem 6.1. Assume that Hypotheses 1 to 6 hold and fix x ∈ Cθ(D), for some θ > 0, andy ∈ E. Then, for any T > 0 and η > 0

limε→0

P

(sup

t∈ [0,T ]|uε − u(t)|E > η

)= 0, (6.1)

where u is the solution of the averaged equation (5.10).

Proof. For any h ∈ D(A1), the slow motion uε satisfies the identity∫Duε(t, ξ)h(ξ) dξ =

∫Dx(ξ)h(ξ) dξ +

∫ t

0

∫Duε(s, ξ)A1h(ξ) dξ ds

+

∫ t

0

∫DB(uε(s, ·))(ξ)h(ξ) dξ ds+

∫ t

0

∫D

[G1(uε(s)h](ξ)dwQ2(s, ξ) +Rε(t),

where

Rε(t) =

∫ t

0

∫D

(B1(uε(s), vε(s))(ξ)− B(uε(s))(ξ)

)h(ξ) dξ ds.

In order to prove Lemma 6.1, we shall need the following result, whose proof is postponed.

Lemma 6.2. Under the same hypotheses of Theorem 6.1, for any T > 0 we have

limε→0

E supt∈ [0,T ]

|Rε(t)|E = 0. (6.2)

Once we have the key result of Lemma 6.2, we proceed exactly as in [7, proof of Theorem6.4]. For the reader’s convenience, we repeat the main steps.

In view of Theorem 3.4, the family L(uε)ε∈ (0,1] is tight in P(C([0, T ];E)). Hence, forany two sequences εnn∈N and εmm∈N converging to zero, we can find two subsequencesεn(k)k∈N and εm(k)k∈N and a sequence

Xkk∈N = (u1,k, u2,k, wk) ⊂ C := [C([0, T ];E)]2 × C([0, T ];D′(D)),

defined on some probability space (Ω, F , P), which converges P-a.s. to some X = (u1, u2, w) ∈C and such that

L(Xk) = (uεn(k) , uεm(k), wQ1), k ∈ N. (6.3)

30

Next, for k ∈ N and i = 1, 2, we define

Ri,k(t) :=

∫Dui,k(t, ξ)h(ξ) dξ −

∫Dx(ξ)h(ξ) dξ −

∫ t

0

∫Dui,k(s, ξ)A1h(ξ) dξ ds

−∫ t

0

∫DB(ui,k(s, ·))(ξ)h(ξ) dξ ds−

∫ t

0

∫D

[G1(ui,k(s)h](ξ)dwQ2(s, ξ).

(6.4)

Due to (6.3), we have that L(R1,k) = L(Rεn(k)) and L(R2,k) = L(Rεm(k)) in C([0, T ];E), so

that

limk→∞

E supt∈ [0,T ]

|Ri,k|E = 0.

Moreover, it is possible to prove that for i = 1, 2 the right hand side in (6.4) convergesP-a.s. to ∫

Dui(t, ξ)h(ξ) dξ −

∫Dx(ξ)h(ξ) dξ −

∫ t

0

∫Dui(s, ξ)A1h(ξ) dξ ds

−∫ t

0

∫DB(ui(s, ·))(ξ)h(ξ) dξ ds−

∫ t

0

∫D

[G1(ui(s)h](ξ)dwQ2(s, ξ),

so that both u1 and u2 are mild solutions of the averaged equation (5.10). By uniqueness, thisimplies u1 = u2 and this means that the two sequences L(uεn(k)) and L(uεm(k)

) are both weaklyconvergent to the same limit u, solution of equation (5.10). As shown in [7], due to a generalargument by Gyongy and Krylov, this implies that uε converges in probability to u.

6.1 Proof of Lemma 6.2

For any n ∈ N, we define

bi,n(ξ, σ1, σ2) :=

bi(ξ, σ1, σ2), if |σ1| ≤ n,

bi(ξ, σ1n/|σ1|, σ2), if |σ1| > n,

i = 1, 2.

In correspondence to each bi,n, we denote by Bi,n the corresponding composition operator andwe have

|x|E ≤ n =⇒ Bi,n(x, y) = Bi(x, y), y ∈ E. (6.5)

Notice that the mappings b1,n and b2,n satisfy all conditions in Hypotheses 2 and 3, respectively.For any fixed ξ ∈ D and σ2 ∈ R, the mappings bi,n(ξ, ·, σ2) : R→ R are Lipschitz continuousand, in view of (2.15),

supξ∈ Dσ2∈R

|b2,n(ξ, σ1, σ2)− b2,n(ξ, ρ1, σ2)| ≤ cn |σ1 − ρ1|, σ1, ρ1 ∈ R. (6.6)

31

Moreover, for any n ∈ N we define

g1,n(ξ, σ1) :=

g1(ξ, σ1), if |σ1| ≤ n,

g1(ξ, σ1n/|σ1|), if |σ1| > n,

and

g2,n(ξ, σ1, σ2) :=

g2(ξ, σ1, σ2), if |σ1| ≤ n,

g2(ξ, σ1n/|σ1|, σ2), if |σ1| > n.

The corresponding composition/multiplication operators are denoted by G1,n and G2,n.

Now, for any n ∈ N we introduce the systemdu(t) = [A1u(t) +B1,n(u(t), v(t))] dt+G1,n(u(t)) dwQ1(t),

dv(t) =1

ε[A2v(t) +B2,n(u(t), v(t))] dt+

1√εG2,n(u(t), v(t)) dwQ2(t),

(6.7)

with initial conditions u(0) = x and v(0) = y. We denote by (uε,n, vε,n) its solution. It isimportant to stress that, as b1,n satisfies Hypothesis 2 and b2,n satisfies Hypothesis 3, then uε,nand vε,n satisfy estimates (3.2), (3.3), (3.7) and (3.10).

Next, for any n ∈ N we introduce the problem

dv(t) = [A2v(t) +B2,n(x, v(t))] dt+G2,n(x, v(t)) dwQ2(t), v(0) = y, (6.8)

whose solution will be denoted by vx,yn . Thanks to (6.5), for any t ≥ 0 we have

vx,yn (t) =

vx,y(t), if |x|E ≤ n,

vxn/|x|E ,y(t), if |x|E > n.

(6.9)

This implies that vx,yn satisfies Hypothesis 5 and a stronger version of Hypothesis 6, namely

lim supT→∞

1

T

∫ T

0E |vx1,0n (t)− vx2,0n (t)|E dt ≤ cn |x1 − x2|E , x1, x2 ∈ E. (6.10)

In particular, for each x ∈ E there exists a unique invariant measure µxn for equation (6.8) andµxn is given by

µxn =

µx, if |x|E ≤ n,

µxn/|x|E , if |x|E > n.

.

Moreover, due to (4.5), for any T > 0 and p ≥ 1 we have

E |vx,yn (t)|pE ≤ cp,n(

1 + e−δpt|y|pE). (6.11)

32

As vx,yn satisfies both Hypothesis 5 and Hypothesis 6, we have that a result analogous toLemma 5.1 holds. More precisely, if we define

Bn(x) =

∫EB1,n(x, y)µxn(dy),

we have that

E∣∣∣∣ 1

T

∫ t+T

t〈B1,n(x, vx,yn (s)),Λ〉E ds−

⟨Bn(x),Λ

⟩E

∣∣∣∣ ≤ α(T )(1 + |x|κ1E + |y|κ2E

)|Λ|E? . (6.12)

Notice that

|x|E ≤ n =⇒ Bn(x) = B(x).

Moreover, as Bn satisfies (6.12), thanks to (6.6), (6.10) and (6.11) it is immediate to checkthat Bn : E → E is globally Lipschitz-continuous.

As in [7], we prove the validity of Lemma 6.2 by using the Khasminskii approach based ontime discretization introduced in [15].

To this purpose, for any ε > 0 we divide the interval [0, T ] in subintervals of size δε > 0,for some constant δε to be determined, and we introduce the auxiliary fast motion vε,n definedin each time interval [kδε, (k + 1)δε], for k = 0, 1, . . . , [T/δε], as the solution of the problem

dv(t) =1

ε[(A2 − α)v(t) +B2,n(uε,n(kδε), v(t))] dt+

1√εG2,n(uε,n(kδε), v(t)) dwQ2(t),

v(kδε) = vε,n(kδε).(6.13)

Notice that, due to the way vε,n has been defined, we have that an estimate analogous to (3.3)holds, that is for any p ≥ 1

supt∈ [0,T ]

E |vε,n(t)|pE ≤ cp,T(1 + |x|pE + |y|pE

). (6.14)

As in [15] and [7], we want to prove the following approximation result.

Lemma 6.3. Assume Hypotheses 1 to 4 and fix x ∈ Cθ(D) and y ∈ E. Then, there exists aconstant κ > 0 such that if

δε = ε log ε−κ,

then for any fixed n ∈ N and p ≥ 1

limε→0

supt∈ [0,T ]

E|vε,n(t)− vε,n(t)|pH = 0. (6.15)

Proof. Let ε > 0 and n ∈ N be fixed. For k = 0, . . . , [T/δε] and t ∈ [kδε, (k + 1)δε], let Γε,n(t)be the solution of the problem

dΓε,n(t) =1

ε(A2 − α)Γε,n(t) dt+

1√εKε,n(t) dwQ2 , Γε,n(kδε) = 0,

33

where

Kε,n(t) := G2,n(uε,n(kδε), vε,n(t))−G2,n(uε,n(t), vε,n(t)).

If we define ρε,n(t) := vε,n(t)− vε,n(t) and zε,n(t) := ρε,n(t)− Γε,n(t), we have

dzε,n(t) =1

ε[(A2 − α)zε,n(t) +Hε,n(t)] dt, zε,n(kδε) = 0,

where

Hε,n(t) := B2,n(uε,n(kδε), vε,n(t))−B2,n(uε,n(t), vε,n(t)).

Notice that, in view of (2.17) we have

Hε,n(t) = B2,n(uε,n(kδε), vε,n(t))−B2,n(uε,n(t), vε,n(t))

−λ(·, uε,n(t), vε,n(t), vε,n(t)) ρε,n(t).

Thanks to (6.6), this yields

d

dt

−|zε,n(t)|E ≤

1

ε

⟨(A2 − α)zε,n(t), δzε,n(t)

⟩E

+1

ε

⟨B2,n(uε,n(kδε), vε,n(t))−B2,n(uε,n(t), vε,n(t)), δzε,n(t)

⟩E

−1

ε

⟨λ(·, uε,n(t), vε,n(t), vε,n(t))zε,n(t), δzε,n(t)

⟩E

−1

ε

⟨λ(·, uε,n(t), vε,n(t), vε,n(t))Γε,n(t), δzε,n(t)

⟩E

≤ −αε|zε,n(t)|E +

cnε|uε,n(kδε)− uε,n(t)|E −

λε,n(t)

ε|zε,n(t)|E +

λε,n(t)

ε|Γε,n(t)|E ,

where

λε,n(t) := λ(ξε,n(t), uε,n(t, ξε,n(t)), vε,n(t, ξε,n(t)), vε,n(t, ξε,n(t))),

and ξε,n(t) is the point in D such that

|zε,n(t, ξε,n(t))| = |zε,n(t)|E .

Therefore, by comparison

|zε,n(t)|E ≤cnε

∫ t

kδε

e−αε

(t−s)|uε,n(kδε)− uε,n(s)|E ds

+1

ε

∫ t

kδε

exp

(−1

ε

∫ t

sλε,n(r) dr

)λε,n(s) |Γε,n(s)|E ds,

34

and, due to (3.10), this yields for any p ≥ 1

E |vε,n(t)− vε,n(t)|pE ≤ cp E |Γε,n(t)|pE + cp,n

(1 + |x|pm2

Cθ(D)+ |y|pE

)δγ(θ)pε

+cp E sups∈ [kδε,t]

|Γε,n(s)|pE

(1

ε

∫ t

kδε

exp

(−1

ε

∫ t

sλε,n(r) dr

)λε,n(s) ds

)p

≤ cp,n(

1 + |x|pm2

Cθ(D)+ |y|pE

)δγ(θ)pε + cp E sup

s∈ [kδε,t]|Γε,n(s)|pE .

(6.16)

Now, by using a factorization argument, for s ∈ [kδε, (k + 1)δε] and η ∈ (0, 1) we have

Γε,n(s) =sinπη

π

1√ε

∫ s

kδε

(s− r)η−1e(s−r) (A2−α)ε Yη,ε,n(r) dr,

where

Yη,ε,n(r) =

∫ r

kδε

(r − ρ)−ηe(r−ρ)(A2−α)

ε Kε,n(ρ) dwQ2(ρ).

Hence, for δ > d/p, that is η > (d+ 2)/2p, due to (2.6) we have

|Γε,n(s)|pE ≤ cp |Γε,n(s)|pδ,p

≤ cη

εp2

∫ s

kδε

[(s− r)η−1

(s− rε

)− δ2

e−α(s−r)

ε

] pp−1

dr

p−1 ∫ s

kδε

|Yη,ε,n(r)|pp dr

≤ cη ε−p2 εp−1ε(η−1)p

∫ s

kδε

|Yη,ε,n(r)|pp dr = cη ε− p

2+ηp−1

∫ s

kδε

|Yη,ε,n(r)|pp dr.

(6.17)

Now, by proceeding as in [4, proof of Theorem 4.2] and [7, proof of Proposition 4.2], we have

E |Yη,ε,n(r)|pp ≤ cpE

∫ r

kδε

(r − ρ)−2η

(r − ρε

)−β2(ρ2−2)ρ2

e−α ρ2+2

ρ2

(r−ρ)ε

[|uε,n(kδε)− uε,n(ρ)|2E + |vε,n(ρ)− vε,n(ρ)|2E

]dr) p

2 .

In view of (2.5), there exists p large enough such that there exists η ∈ (0, 1) with

d+ 2

p< 2η < 1− β2(ρ2 − 2)

ρ2.

Then

E∫ s

kδε

|Yη,ε,n(r)|pp dr

≤ cp ε−(ηp+ p2 )∫ s

kδε

(E |uε,n(kδε)− uε,n(r)|pE + E |vε,n(r)− vε,n(r)|pE

)dr.

35

In view of (6.17) and (3.10), this yields

E sups∈ [kδε,t]

|Γε,n(s)|pE ≤ cη,p(

1 + |x|pm2

Cθ(D)+ |y|pE

)δγ(θ)pε ζε

+cη,p1

ε

∫ t

kδε

E |vε,n(s)− vε,n(s)|pE ds,

so that, thanks to (6.16), for t ∈ [kδε, (k + 1)δε]

E |vε,n(t)− vε,n(t)|pE ≤ cη,p(

1 + |x|pm2

Cθ(D)+ |y|pE

)δγ(θ)pε (1 + ζε)

+cpε

∫ t

kδε

E |vε,n(s)− vε,n(s)|pE ds.

From the Gronwall Lemma, this gives

E |vε,n(t)− vε,n(t)|pE ≤ cη,p(

1 + |x|pm2

Cθ(D)+ |y|pE

)δγ(θ)pε (1 + ζε) e

ζε .

Now, if

eζε = exp

(log

1

εκ

)=

1

εκ,

we have

δγ(θ)pε (1 + ζε) e

ζε =1

εκδγ(θ)pε

(1 + log

1

εκ

)=

(log

1

εκ

)κ(1 + log

1

εκ

)δγ(θ)p−κε .

Hence, if we take κ < γ(θ), we have (6.15) for any p ≥ 1.

Finally, we can prove (6.2). For any n ∈ N, we have

E supt∈ [0,T ]

|Rε(t)| = E

(sup

t∈ [0,T ]|Rε(t)|; sup

t∈ [0,T ]|uε(t)|E ≤ n

)

+E

(sup

t∈ [0,T ]|Rε(t)|; sup

t∈ [0,T ]|uε(t)|E > n

)

≤ E

(sup

t∈ [0,T ]|Rε,n(t)|

)+ E

(sup

t∈ [0,T ]|Rε(t)|2E

) 12

P

(sup

t∈ [0,T ]|uε(t)|E > n

) 12

,

(6.18)

where

Rε,n(t) =

∫ t

0

∫D

(B1,n(uε,n(s), vε,n(s))(ξ)− Bn(uε,n(s))(ξ)

)h(ξ) dξ ds.

Now, according to (2.22) and (5.2), we have

|Rε(t)| ≤ c |h|E∫ t

0

(1 + |uε(s)|m1

E + |vε(s)|E)ds,

36

so that, thanks to (3.2) and (3.3) we obtain

E supt∈ [0,T ]

|Rε(t)|2 ≤ cT(

1 + |x|2m1E + |y|2E

)|h|2E .

By using again estimate (3.2), due to (6.18) this yields

E supt∈ [0,T ]

|Rε(t)| ≤ E

(sup

t∈ [0,T ]|Rε,n(t)|

)+cTn

(1 + |x|2m1

E + |y|2E)|h|E .

Therefore, due to the arbitrariness on n ∈ N, (6.2) follows once we prove that for any fixedn ∈ N

limε→0

E

(sup

t∈ [0,T ]|Rε,n(t)|

)= 0. (6.19)

In view of key Lemma 6.3, of estimates (3.2) and (3.3) and of the Lipschitz continuity ofB1,n(ξ, ·, y), for any fixed ξ ∈ D and y ∈ E, and of Bn, the proof of (6.19) is analogous to [7,proof of Lemma 6.3]. Here we repeat it for the reader convenience.

We have

lim supε→0

E supt∈ [0,T ]

∣∣∣∣∫ t

0

⟨B1,n(uε,n(s), vε,n(s))− Bn(uε,n(s)), h

⟩Hds

∣∣∣∣≤ lim sup

ε→0E∫ T

0

∣∣〈B1,n(uε,n(s), vε,n(s))−B1,n(uε,n([s/δε]δε), vε,n(s)), h〉H∣∣ ds

+ lim supε→0

E supt∈ [0,T ]

∣∣∣∣∫ t

0

⟨B1,n(uε,n([s/δε]δε), vε,n(s))− Bn(uε,n(s)), h

⟩Hds

∣∣∣∣ .(6.20)

Now, due to (2.27) and (6.6) we have

E∫ T

0

∣∣〈B1,n(uε,n(s), vε,n(s))−B1,n(uε,n([s/δε]δε), vε,n(s)), h〉H∣∣ ds

≤ cn |h|H∫ T

0E |uε,n(s)− uε,n([s/δε]δε)|E ds

+c |h|H∫ T

0E |vε,n(s)− vε,n(s)|E

(1 + |uε,n(s)|θE + |vε,n(s)|θE + |vε,n(s)|θE

)ds

≤ cn |h|H∫ T

0E |uε,n(s)− uε,n([s/δε]δε)|E ds+ c |h|H sup

t∈ [0,T ]

(E |vε,n(t)− vε,n(t)|2E

) 12

×∫ T

0

(E(

1 + |uε,n(s)|2θE + |vε,n(s)|2θE + |vε,n(s)|2θE)) 1

2ds.

37

and then, thanks to (3.2), (3.3), (3.10) and (6.14), we conclude

E∫ T

0

∣∣〈B1,n(uε,n(s), vε,n(s))−B1,n(uε,n([s/δε]δε), vε,n(s)), h〉H∣∣ ds

≤ cT,n |h|H(

1 + |x|(2∨θ)m1

Cθ(D)+ |y|2∨θE

)(δγ(θ)ε + sup

t∈ [0,T ]

(E |vε,n(t)− vε,n(t)|2E

) 12

).

Therefore, in view of Lemma 6.3, from (6.20)

lim supε→0

E supt∈ [0,T ]

∣∣∣∣∫ t

0

⟨B1,n(uε,n(s), vε,n(s))− Bn(uε,n(s)), h

⟩Hds

∣∣∣∣= lim sup

ε→0E supt∈ [0,T ]

∣∣∣∣∫ t

0

⟨B1,n(uε,n([s/δε]δε), vε,n(s))− Bn(uε,n(s)), h

⟩Hds

∣∣∣∣ .Now, we have

E supt∈ [0,T ]

∣∣∣∣∫ t

0

⟨B1,n(uε,n([s/δε]δε), vε,n(s))− Bn(uε,n(s)), h

⟩Hds

∣∣∣∣≤

[T/δε]∑k=0

E

∣∣∣∣∣∫ (k+1)δε

kδε

⟨B1,n(uε,n([s/δε]δε), vε,n(s))− Bn(uε,n(kδε)), h

⟩Hds

∣∣∣∣∣+

[T/δε]∑k=0

∫ (k+1)δε

kδε

E∣∣⟨Bn(uε,n(kδε))− Bn(uε,n(s), h

⟩H

∣∣ ds,and then, due to (3.10) and to the global Lipschitz-continuity of Bn

E supt∈ [0,T ]

∣∣∣∣∫ t

0

⟨B1,n(uε,n([s/δε]δε), vε,n(s))− Bn(uε,n(s)), h

⟩Hds

∣∣∣∣≤

[T/δε]∑k=0

E

∣∣∣∣∣∫ (k+1)δε

kδε

⟨B1,n(uε,n([s/δε]δε), vε,n(s))− Bn(uε,n(kδε)), h

⟩Hds

∣∣∣∣∣+cT,n |h|H

(1 + |x|m1

Cθ(D)+ |y|E

)[T/δε] δ

γ(θ)+1ε .

This means that in order to obtain (6.19) and conclude the proof of Lemma 6.2, it remains toshow that

limε→0

[T/δε]∑k=0

E

∣∣∣∣∣∫ (k+1)δε

kδε

⟨B1,n(uε,n([s/δε]δε), vε,n(s))− Bn(uε,n(kδε)), h

⟩Hds

∣∣∣∣∣ = 0. (6.21)

38

We have

E

∣∣∣∣∣∫ (k+1)δε

kδε

⟨B1,n(uε,n([s/δε]δε), vε,n(s))− Bn(uε,n(kδε)), h

⟩Hds

∣∣∣∣∣= E

∣∣∣∣∫ δε

0

⟨B1,n(uε,n([s/δε]δε), vε,n(kδεs))− Bn(uε,n(kδε)), h

⟩Hds

∣∣∣∣= E

∣∣∣∣∫ δε

0

⟨B1,n(uε,n([s/δε]δε), v

uε,n(kδε),vε,n(kδε)n (s/ε)− Bn(uε,n(kδε)), h

⟩Hds

∣∣∣∣= δεE

∣∣∣∣ 1

ζε

∫ ζε

0

⟨B1,n(uε,n([s/δε]δε), v

uε,n(kδε),vε,n(kδε)n (s)− Bn(uε,n(kδε)), h

⟩Hds

∣∣∣∣ ,where v

uε,n(kδε),vε,n(kδε)n (s) is the solution of the fast motion equation (4.1), with frozen slow

component uε,n(kδε) and initial datum vε,n(kδε) and noise wQ2 independent of both of them.According to (6.12), (3.2) and (3.3), this yields

E

∣∣∣∣∣∫ (k+1)δε

kδε

⟨B1,n(uε,n([s/δε]δε), vε,n(s))− Bn(uε,n(kδε)), h

⟩Hds

∣∣∣∣∣≤ δεα(ζε)

(1 + E |uε,n(kδε)|κ1E + E |vε,n(kδε)|κ2E

)|h|1

≤ dεα(ζε)(1 + |x|κ1E + |y|κ2E

)|h|1,

and (6.21) follows.

References

[1] V. I. Arnold, V. V. Kozlov, A. I. Neishtadt, Mathematical aspects of classicaland celestial mechanics. [Dynamical systems. III], third edition, Encyclopaediaof Mathematical Sciences, Springer-Verlag, Berlin, 2006.

[2] N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic methods in the theory of non-linear oscillations, Gordon and Breach Science Publishers, New York 1961.

[3] S. Cerrai, Second order PDE’s in finite and infinite dimension. A probabilisticapproach, Lecture Notes in Mathematics Series 1762, Springer Verlag (2001).

[4] S. Cerrai, Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitzreaction term, Probability Theory and Related Fields 125 (2003), pp. 271-304.

[5] S. Cerrai, Asymptotic behavior of systems of SPDE’s with multiplicative noise, StochasticPartial Differential Equations and Applications VII, Lecture Notes in Pure and AppliedMathematics 245 (2005), Chapman and Hall/CRC Press, pp. 61-75.

39

[6] S. Cerrai, M. Freidlin, Averaging principle for a class of SPDE’s, Probability Theory andRelated Fields 144 (2009), pp. 137-177.

[7] S. Cerrai, A Khasminskii type averaging principle for stochastic reaction-diffusion equa-tions, Annals of Applied Probability 19 (2009), pp. 899-948.

[8] S. Cerrai, Normal deviations from the averaged motion for some reaction-diffusion equa-tions with fast oscillating perturbation, Journal des Mathematiques Pures et Appliquees91 (2009), pp. 614-647.

[9] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, CambridgeUniversity Press, Cambridge (1992).

[10] G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems, LondonMathematical Society, Lecture Notes Series 229, Cambridge University Press, Cambridge(1996).

[11] M. I. Freidlin, On stable oscillations and equilibrium induced by small noise, Journal ofStatistical Physics 103 (2001), pp. 283-300.

[12] M. I. Freidlin, A. D. Wentzell, Random Perturbations of Dynamical Systems,Second Edition, Springer Verlag (1998).

[13] M. I. Freidlin, A. D. Wentzell, Long-time behavior of weakly coupled oscillators, Journalof Statistical Physics 123 (2006), pp. 1311-1337.

[14] I. Gyongy, N.V. Krylov, Existence of strong solutions for Ito’s stochastic equations viaapproximations, Probability Theory and Related Fields 103 (1996), pp. 143-158.

[15] R. Z. Khasminskii, On the principle of averaging the Ito’s stochastic differential equations(Russian), Kibernetika 4 (1968), pp. 260-279.

[16] Y. Kifer, Some recent advances in averaging, Modern dynamical systems and applications,pp. 385-403, Cambridge Univ. Press, Cambridge, 2004.

[17] Y. Kifer, Diffusion approximation for slow motion in fully coupled averaging, ProbabilityTheory and Related Fields 129 (2004), pp. 157-181.

[18] Y. Kifer, Averaging and climate models, Stochastic climate models (Chorin, 1999), pp.171-188, Progress in Probability 49, Birkhuser, Basel, 2001.

[19] Y. Kifer, Stochastic versions of Anosov’s and Neistadt’s theorems on averaging, Stochas-tics and Dynamics 1 (2001), pp. 1-21.

[20] S. B. Kuksin, A. L. Piatnitski, Khasminski-Whitman averaging for randonly perturbedKdV equations, Preprint (2006).

[21] B. Maslowski, J. Seidler, I. Vrkoc, An averaging principle for stochastic evolution equa-tions. II, Mathematica Bohemica 116 (1991), pp. 191-224.

40

[22] A. I. Neishtadt, Averaging in multyfrequency systems, Soviet Physics Doktagy 21 (1976),pp. 80-82.

[23] G. C. Papanicolaou, D. Stroock, S. R. S. Varadhan, Martingale approach to some limit the-orems, in Papers from the Duke Turbolence Conference (Duke Univ. Duhram,N.C. 1976), Paper 6, ii+120 pp. Duke Univ. Math. Ser. Vol. III, Duke Univ. Duhram,N.C. (1977).

[24] A. J. Roberts, W. Wang, Average and deviation for slow-fast stochastic partial differentialequations, arXiv:0904.1462v1, 2009.

[25] J. Seidler, I. Vrkoc, An averaging principle for stochastic evolution equations, CasopisPest. Mat. 115 (1990), pp. 240-263.

[26] H. C. Tuckwell, Random perturbations of the reduced Fitzhugh-Nagumo equation, PhysicaScripta 46, pp. 481-484.

[27] A. Y. Veretennikov, On the averaging principle for systems of stochastic differential equa-tion, Mathematics of the USSR-Sbornik 69 (1991), pp. 271-284.

[28] V. M. Volosov, Averaging in systems of ordinary differential equations, Russian Matehmat-ical Surveys 17 (1962), pp. 1-126.

41