Derivative Formula, Coupling Property and Strong Feller ...pkim/7ICSAA/Slides/Zhao_Dong.pdf · Z....

Derivative Formula, Coupling Property and Strong Fellerfor S(P)DEs Driven by Levy Processes

Z. DongJoint work with: X.H. Peng, Y. L. Song, X.C. Zhang

Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences

7th International Conference on Stochastic Analysis and itsApplications, Seoul national university, August 6-11, 2014

Z. Dong (AMSS CAS ) Derivative Formula, Coupling and Strong Feller 28-04-2014 1 / 42

Outline

1 Introduction: Framework and Known Results

2 Derivative formula and coupling property for SDEs

3 Derivative formula and coupling property for SPDEs

4 Strong Feller Property for SDEs driven by degenerate additive noise

5 Classical Wiener Space

Framework

The Bismut formula (also called Bismut-Elworthy-Li formula) is afundamental tool in stochastic analysis. Let, for instance, Xt be adiffusion process on Rn generated by an elliptic differential operator andPtt≥0 be the associated Markov semigroup. The Bismut formula is oftype

∇ξPt f (x) = Ef (X xt )Mx

t , f ∈ Bb(Rn), t > 0,

where Mxt is a random variable independent of f , and ∇ξ is the directional

derivative along ξ.

Framework

Applications of Derivative Formula:

Strong Feller properties

Heat kernel estimates

Functional inequalities · · · · · ·

Framework

Functional inequalities

· · · · · ·

Framework

Development:

Diffusion (jump-diffusion) case:♣ Non-degenerate Wiener processes: J.M. Bismut( Large Deviations and theMalliavin Calculus, Birkhauser, Boston, 1984), K.D. Elworthy and X.M.Li(JFA,1994), S. Peszat and J. Zabczyk(Ann.Prob,1995), A. Takeuchi(JTheor.Prob.,2010), Z.Dong and Y.C.Xie(JDE,2011), B. Xie(Poten. Ana., 2012)· · ·♣ Degenerate Wiener processes: A. Guillin and F.Y. Wang(JDE,2012), F.Y. Wang

and X.C. Zhang(JMPA,2013), · · ·

Purely jump case:♣ Non-degenerate jump processes: R.F. Bass and M. Cranston. (Ann. Probab.,1986), J.R. Norris, (Seminaire de Probabilites XXII. Lect. Notes. Math., 1988),Z.Q.Cheng(2010),F.Y. Wang(SPA, 2012), X.C. Zhang(SPA, 2012), E. Priola andJ. Zabczyk.(PTRF, 2011), J. Wang and F.Y. Wang(SPA, 2012)· · ·♣ Degenerate jump processes: Few results

Framework

Development:

Diffusion (jump-diffusion) case:♣ Non-degenerate Wiener processes: J.M. Bismut( Large Deviations and theMalliavin Calculus, Birkhauser, Boston, 1984), K.D. Elworthy and X.M.Li(JFA,1994), S. Peszat and J. Zabczyk(Ann.Prob,1995), A. Takeuchi(JTheor.Prob.,2010), Z.Dong and Y.C.Xie(JDE,2011), B. Xie(Poten. Ana., 2012)· · ·♣ Degenerate Wiener processes: A. Guillin and F.Y. Wang(JDE,2012), F.Y. Wang

and X.C. Zhang(JMPA,2013), · · ·

Purely jump case:♣ Non-degenerate jump processes: R.F. Bass and M. Cranston. (Ann. Probab.,1986), J.R. Norris, (Seminaire de Probabilites XXII. Lect. Notes. Math., 1988),Z.Q.Cheng(2010),F.Y. Wang(SPA, 2012), X.C. Zhang(SPA, 2012), E. Priola andJ. Zabczyk.(PTRF, 2011), J. Wang and F.Y. Wang(SPA, 2012)· · ·♣ Degenerate jump processes: Few results

Recall two Facts

Coupling Property:(Cranston, Greven.,1995,SPA)A strong Markov process on a Polish space has coupling property if andonly if

limt→∞

‖Pt(x , ·)− Pt(y , ·)‖Var = 0, x , y ∈ Rn

where Pt(x , ·) is the transition probability and ‖ · ‖Var denotes the totalvariation norm.

Strong Feller:(Da Prato, Zabczyk, Erg.Inf.Dimen.Sys.,1995) A Markovsemigroup Pt on Bb(Rn) is strong Feller if ∀f ∈ C 2

b (Rn), one has

|Pt f (x)− Pt f (y)| ≤ C‖f ‖∞|x − y |.

Recall two Facts

Coupling Property:(Cranston, Greven.,1995,SPA)A strong Markov process on a Polish space has coupling property if andonly if

limt→∞

‖Pt(x , ·)− Pt(y , ·)‖Var = 0, x , y ∈ Rn

where Pt(x , ·) is the transition probability and ‖ · ‖Var denotes the totalvariation norm.

Strong Feller:(Da Prato, Zabczyk, Erg.Inf.Dimen.Sys.,1995) A Markovsemigroup Pt on Bb(Rn) is strong Feller if ∀f ∈ C 2

b (Rn), one has

|Pt f (x)− Pt f (y)| ≤ C‖f ‖∞|x − y |.

Outline

Framework

Consider following semilinear SDEs:dXt = b(Xt)dt + σtdLt ,

X0 = x ,(1)

where b : Rn → Rn and σ : [0,∞)→ Rn ⊗ Rn are measurable. L is aLevy process with characteristic measure ν.

Hypothesis

(H2.1) There exists a differentiable function ρ : Rn0 → (0,∞) satisfying

ν(dz) ≥ ν1(dz) := ρ(z)dz .

(H2.2) b ∈ C 1(Rn) with ∇b bounded and Lipschitz continuous. Andthere exists a constant β > 0 such that |σ−1

s | ≤ β for any s > 0.

(H2.3) There is a constant K > 0 such that

〈b(x)− b(y), x − y〉 ≤ −K |x − y |2

for any x , y ∈ Rn.

Hypothesis

ν(dz) ≥ ν1(dz) := ρ(z)dz .

〈b(x)− b(y), x − y〉 ≤ −K |x − y |2

Hypothesis

ν(dz) ≥ ν1(dz) := ρ(z)dz .

〈b(x)− b(y), x − y〉 ≤ −K |x − y |2

Framework

Due to (H2.1), L can be decomposed into two independent parts:

Lt = L1t + L2

where L1 is purely jump process with ν1(dz). The jump measure of L1 isdenoted by N(dz , dt).

Framework

If λ := ν1(Rn0) =∞, we aim to investigate the Bismut type formula for

Pt f (x) := Ef (X xt )

P1t f (x) := E

f (X x

t )I[Nt≥1]

where t ≥ 0, x ∈ Rn, f ∈ Bb(Rn),Nt := N([0, t]× Rn).

Framework

If λ := ν1(Rn0) =∞, we aim to investigate the Bismut type formula for

Pt f (x) := Ef (X xt )

P1t f (x) := E

f (X x

t )I[Nt≥1]

where t ≥ 0, x ∈ Rn, f ∈ Bb(Rn),Nt := N([0, t]× Rn).

L1-derivative

For T > 0, let V = V (s, z)s≤T ,z∈Rn0

be a predictable and integrableprocess. Let ε > 0, define

Nε(B × [0, t]) =

IB(z + εV (s, z))N(dz , ds),

where ν(B) <∞.

Definition (Bass,1986, Ann. Prob.)

A functional Ft(N) := F (N(dz , ds)|s≤t) is called to have anL1-derivative in the direction V , if there exists an integrable randomvariable denoted by DVFt(N), such that

limε→0

E∣∣Ft(N

ε)− Ft(N)

ε− DVFt(N)

∣∣ = 0.

L1-derivative

For T > 0, let V = V (s, z)s≤T ,z∈Rn0

be a predictable and integrableprocess. Let ε > 0, define

Nε(B × [0, t]) =

IB(z + εV (s, z))N(dz , ds),

where ν(B) <∞.

Definition (Bass,1986, Ann. Prob.)

A functional Ft(N) := F (N(dz , ds)|s≤t) is called to have anL1-derivative in the direction V , if there exists an integrable randomvariable denoted by DVFt(N), such that

limε→0

E∣∣Ft(N

ε)− Ft(N)

ε− DVFt(N)

∣∣ = 0.

Integration by Parts Formula

Denote

V =V : Ω× [0,T ]× Rn

0 → Rn∣∣∣V is predictable with V and DzV bounded,

∃U0 ⊂ Rn0 compact, s.t. SuppV ⊂ [0,T ]× U0.

Proposition 2.1(Norris,1988)

Let (H2.1) with ρ ∈ C 1(Rn0). If a functional Ft(N) has an L1-derivative

DVFt(N) for V ∈ V, then

EDVFt(N) = −EFt(N)Rt,

div(ρ(z)V (s, z))

ρ(z)N(dz , ds).

Derivative formula for P1t

Let Jt be the derivative of X xt w.r.t. the initial value x .

Theorem 2.2(Dong, Song, 2013.)

Let (H2.1)-(H2.2) hold and ρ ∈ C 1(Rn) with∫Rn |∇ρ(z)|dz <∞. For

t > 0, ξ ∈ Rn and f ∈ Cb(Rn), we have

∇ξP1t f (x) = −E

f (X x

t )I[Nt≥1]

∇ log ρ(z) · (σ−1s Jsξ)N(dz , ds)

Furthermore,

‖∇P1t f ‖∞ ≤

4βet‖∇b‖∞

λ(1− e−λt − e−λtλ0t)‖f ‖∞

|∇ρ(z)|dz .

Let (H2.1)-(H2.3) hold and ρ ∈ C 1(Rn) with∫Rn |∇ρ(z)|dz <∞. Then

for t > K+λK ,

‖Pt(x , ·)− Pt(y , ·)‖Var ≤ 4β

|∇ρ(z)|dz |x − y |+ 2e−

KλK+λ

Remark: From Theorem 2.2 or Theorem 2.3, we can not obtain the strongFeller of Pt under the condition λ <∞. The reason is that with a positiveprobability the process does not jump before a fixed time t > 0. But thecoupling property of Pt is investigated.

Notations

Denote

Li =h : Rn → R|

|h(z)|iν1(dz) <∞, i = 1, 2;

C =h : Rn → R

∣∣∣h is differential and has compact support in Rn0

For h ∈ C , we define a weighted norm as

‖h‖ρ =∫

|∇h(z)|2ν1(dz) 1

h2(z)|∇ log ρ(z)|2ν1(dz) 1

Let C‖·‖ρ

be the closure of C under ‖ · ‖ρ. Denote

Hρ =h ∈ L1 ∩ L2

∣∣∣h ≥ 0, h ∈ C‖·‖ρ

and ∇h is bounded..

Notations

Denote

Li =h : Rn → R|

|h(z)|iν1(dz) <∞, i = 1, 2;

C =h : Rn → R

‖h‖ρ =∫

|∇h(z)|2ν1(dz) 1

h2(z)|∇ log ρ(z)|2ν1(dz) 1

Let C‖·‖ρ

Hρ =h ∈ L1 ∩ L2

∣∣∣h ≥ 0, h ∈ C‖·‖ρ

Notations

Denote

Li =h : Rn → R|

|h(z)|iν1(dz) <∞, i = 1, 2;

C =h : Rn → R

‖h‖ρ =∫

|∇h(z)|2ν1(dz) 1

h2(z)|∇ log ρ(z)|2ν1(dz) 1

Let C‖·‖ρ

Hρ =h ∈ L1 ∩ L2

∣∣∣h ≥ 0, h ∈ C‖·‖ρ

Notations

Denote

Li =h : Rn → R|

|h(z)|iν1(dz) <∞, i = 1, 2;

C =h : Rn → R

‖h‖ρ =∫

|∇h(z)|2ν1(dz) 1

h2(z)|∇ log ρ(z)|2ν1(dz) 1

Let C‖·‖ρ

Hρ =h ∈ L1 ∩ L2

∣∣∣h ≥ 0, h ∈ C‖·‖ρ

Derivative Formula for Pt

Let (H2.1)-(H2.2) hold and ρ ∈ C 1(Rn0). If θ := lim inf

x→∞ν1([h≥x−1])

log x > 0 for

some h ∈ Hρ, then for t > 8(θ∧1)(1−e−1)

, ξ ∈ Rn and f ∈ Cb(Rn),

∇ξPt f (x) =− E

f (X x

t )[H−1

〈∇(ρ(z)h(z))

ρ(z), σ−1

s Jsξ〉N(dz , ds)

+ H−2t

〈∇h(z), σ−1s Jsh(z)ξ〉N(dz , ds)

where Ht =∫ t

0h(z)N(dz , ds).

Continuation of Theorem 2.6

Furthermore,

‖∇Pt f ‖∞ ≤C(

t − 8(θ∧1)(1−e−1)

)e‖∇b‖∞tβ‖f ‖∞

×∥∥ |∇(hρ)|

∥∥L2 + 2‖∇h‖∞

√‖h‖2

L2 + ‖h‖2L1

where C is a constant independent of t.

Remark: The condition θ > 0 can ensure Levy measure ν1 is infinite.Indeed, from Theorem 2.6 and classical approximation argument, we canderive strong Feller property of Pt when b is Lipschitz continuous.

Continuation of Theorem 2.6

Furthermore,

‖∇Pt f ‖∞ ≤C(

t − 8(θ∧1)(1−e−1)

)e‖∇b‖∞tβ‖f ‖∞

×∥∥ |∇(hρ)|

∥∥L2 + 2‖∇h‖∞

√‖h‖2

L2 + ‖h‖2L1

where C is a constant independent of t.

Remark: The condition θ > 0 can ensure Levy measure ν1 is infinite.Indeed, from Theorem 2.6 and classical approximation argument, we canderive strong Feller property of Pt when b is Lipschitz continuous.

SDEs Driven by α-stable Process

As an example, we have

Example

Let (H2.2) hold. If ρ(z) = Cα|z|n+α with Cα > 0 and 0 < α < 2, then for

t ≥ 1 + 81−e−1 , ξ ∈ Rn and f ∈ Cb(Rn),

‖∇ξPt f ‖∞ ≤ C (n, α)‖f ‖∞|ξ|βe‖∇b‖∞t ,

where C (n, α) denotes a constant only depending on n and α.

Outline

Framework

Let (H, 〈·, ·〉) be a separable Hilbert space and µ be a Gaussian measureon H with covariance operator Q.

Quasi-invariant Property:

Under the shift z 7→ z + h for any h ∈ImQ12 , µ(·+ h) and µ are mutually

absolutely continuous.

ϕ(z , h) :=µ(dz + h)

µ(dz)= exp〈h, z〉0 −

2〈h, h〉0, µ− a.s,

where 〈·, ·〉0 stands for the inner product induced by Q12 and equipped on

Framework

Let (H, 〈·, ·〉) be a separable Hilbert space and µ be a Gaussian measureon H with covariance operator Q.

Quasi-invariant Property:

Under the shift z 7→ z + h for any h ∈ImQ12 , µ(·+ h) and µ are mutually

absolutely continuous.

ϕ(z , h) :=µ(dz + h)

µ(dz)= exp〈h, z〉0 −

2〈h, h〉0, µ− a.s,

where 〈·, ·〉0 stands for the inner product induced by Q12 and equipped on

Framework

Consider SDEs on H:dXt = AXtdt + F (Xt)dt + dLt + dZt ,

X0 = x ,(2)

where A : D(A) ⊂ H→ H is an adjoint, unbounded and linear operatorgenerating a C0-semigroup Stt≥0 on H. L := Ltt≥0 is a Levy processon H with Levy measure ν. Z := Ztt≥0 is another square-integrableLevy process independent of L.

Hypothesis

(H3.1) ImS(t) ⊂ ImQ holds for any t > 0.

(H3.2) F : H→ H is Frechet differentiable with ∇F bounded andLipschitz continuous.

(H3.3) A is a dissipative operator defined by

A =∑k≥1

(−γk)ek ⊗ ek , (3)

for 0 < γ1 ≤ γ2 ≤ · · · ≤ γk ≤ · · · and γk →∞ as k →∞.

Hypothesis

A =∑k≥1

(−γk)ek ⊗ ek , (3)

Hypothesis

A =∑k≥1

(−γk)ek ⊗ ek , (3)

(H3.4) There exists a differentiable function ρ : H→ (0,∞) with ∇ρbounded and satisfying

∫Hρ(z)µ(dz) <∞ and

∫H|z |2ρ(z)µ(dz) <∞,

such that

ν(dz) = ρ(z)µ(dz).

V =V : Ω× [0,T ]→ ImQ

∣∣∣V is predictable and

E|Q−1V (s)|ds <∞..

Theorem 3.1(Dong, Song, Xu 2013)

Suppose (H3.4) holds. For V ∈ V and f ∈ C 2b (H),

EDV f (Lt)

= −E

f (Lt)Mt

, t ≤ T , (4)

where Mt =∫ t

(〈z ,Q−1V (s)〉 + 〈∇ log ρ(z),V (s)〉

)N(dz , ds).

V =V : Ω× [0,T ]→ ImQ

∣∣∣V is predictable and

E|Q−1V (s)|ds <∞..

Suppose (H3.4) holds. For V ∈ V and f ∈ C 2b (H),

EDV f (Lt)

= −E

f (Lt)Mt

, t ≤ T , (4)

where Mt =∫ t

(〈z ,Q−1V (s)〉 + 〈∇ log ρ(z),V (s)〉

)N(dz , ds).

Derivative Formula

C 1b (H) =

G : H→ H

∣∣G and its first order derivatives are continuous and bounded..

Let (H3.1), (H3.2) and (H3.4) hold. If∫ t

0 ‖Q−1S(s)‖ds <∞, then for

f ∈ C 1b (H) and ξ ∈ H,

∇ξP1t f (x) = −E

f (X x

t )I[Nt≥1]

(〈z ,Q−1Jsξ〉+ 〈∇ log ρ(z), Jsξ〉

)N(dz , ds)

Example

Let (H3.3) hold. For 0 < δ < 12 , (−A)δ denotes the fractional power of

−A, defined by

(−A)δ =1

Γ(δ)

∫ ∞0

t−δS(t)dt,

where Γ is the Euler function. It can be proved that S(t)H ⊂ D((−A)δ),for any t > 0, and

‖(−A)δS(t)‖ ≤ Cδt−δ

for a suitable positive constant Cδ. Take Q =((−A)δ

)−1, then we have

S(t)H ⊂ ImQ. Moreover,

limt→∞

∫ t0 ‖Q

−1S(s)‖2ds

t≤ lim

t→∞

∫ t0 s−2δds

Outline

Brownian motion

W :=w : [0,∞)→ Rm|w is continuous with w0 = 0.

W is endowed with the locally uniform topology and the probabilitymeasure µW so that the coordinate process

Wt(w) := wt = (w1t , · · · ,wm

is an m-dimensional Brownian motion.

Brownian motion

W :=w : [0,∞)→ Rm|w is continuous with w0 = 0.

W is endowed with the locally uniform topology and the probabilitymeasure µW so that the coordinate process

Wt(w) := wt = (w1t , · · · ,wm

is an m-dimensional Brownian motion.

Subordinator

S :=` : R+ → Rm

+|` is cadlag with `0 = 0, each component

being increasing and purely jumping.

S is endowed with the Skorohod metric and the probability measureµS so that the coordinate process

St(`) := `t = (`1t , · · · , `mt )

is an m-dimensional Levy process with Laplace transform

EµS(e−z·St ) = exp

(e−z·u − 1)νS(du)

Subordinator

S :=` : R+ → Rm

+|` is cadlag with `0 = 0, each component

being increasing and purely jumping.

S is endowed with the Skorohod metric and the probability measureµS so that the coordinate process

St(`) := `t = (`1t , · · · , `mt )

is an m-dimensional Levy process with Laplace transform

EµS(e−z·St ) = exp

(e−z·u − 1)νS(du)

Subordinated Brownian motion

Consider the following product probability space

(Ω,F ,P) :=(W× S,B(W)×B(S), µW × µS

Lift Wt and St to this probability space, then Wt and St are independent,and the subordinated Brownian motion

WSt :=(W 1

S1t, · · · ,Wm

)is an m-dimensional Levy process.

Subordinated Brownian motion

Assume

P(ω ∈ Ω : ∃j = 1, · · · ,m and ∃t > 0 such that S jt (ω) = 0) = 0, (6)

which means that St is nondegenerate along each direction.

Remark

α-stable subordinator meets this assumption.

The SDE driven by subordinated Brownian motion

Consider the following SDE:

dXt = b(Xt)dt + σdWSt , X0 = x , (7)

where b : Rd → Rd is a smooth function, σ is a d ×m matrix.

Hormander’s type condition

Hormander’s type condition at point x ∈ Rd :

∃n = n(x) ∈ N, s.t.

Rank[σ,B1(x)σ,B2(x)σ, · · · ,Bn(x)σ] = d , (8)

where B1(x) := (∇b)ij(x) = (∂jbi (x))ij , and for n ≥ 2,

Bn(x) := (b · ∇)Bn−1(x)− (∇b · Bn−1)(x).

Strong Feller property

Theorem 4.1(Dong, Peng, Song, Zhang 2013)

Assume that

the solution to (7) globally exists

(b, σ) satisfy Hormander’s type condition (8) at each point x ∈ Rd .

Then for any t > 0, the law of Xt(x) is continuous w.r.t. variable x in thetotal variation distance. In particular, the semigroup (Pt)t>0 has thestrong Feller property, i.e., for any t > 0 and f ∈ Bb(Rd),

x 7→ Ef (Xt(x)) is continuous.

Remark

This result has been much extended to multiplicative noise by Xi ChengZhang 2013.

Example

Stochastic oscillators:dzi (t) = ui (t)dt, i = 1, · · · , d ,dui (t) = −∂ziH(z(t), u(t))dt, i = 2, · · · , d − 1,

dui (t) = −[∂ziH(z(t), u(t)) + γiui (t)]dt +√TidW

iS it, i = 1, d ,

where d ≥ 3, γ1, γd ∈ R, T1,Td > 0, and

H(z , u) :=d∑

2|ui |2 + V (zi )

d−1∑i=1

U(zi+1 − zi ).

Example

Proposition 4.2

Assume that V ,U ∈ C∞(R) are nonnegative and lim|z|→∞ V (z) =∞. If

U is strictly convex, then for any f ∈ Bb(Rd × Rd), the map

(z0, u0) 7→ Ez0,u0f (zt , ut)

is continuous.

Outline

Classical Wiener Space

Let W be the space of all continuous functions from R+ := [0,∞) toRm vanishing at starting point 0 and the Wiener measure µW so thatthe coordinate process

Wt(ω) = ωt

is a standard m-dimensional Brownian motion.

Let H ⊂W be the Cameron-Martin space. The inner product in H isdenoted by

〈h1, h2〉H :=m∑i=1

∫ ∞0

hi1(s)hi2(s)ds.

Let D be the Malliavin derivative operator.

Let Dk,p be the associated Wiener-Sobolev space with the norm

‖F‖k,p := ‖F‖p + ‖DF‖p + · · ·+ ‖DkF‖p,

Wt(ω) = ωt

〈h1, h2〉H :=m∑i=1

∫ ∞0

hi1(s)hi2(s)ds.

‖F‖k,p := ‖F‖p + ‖DF‖p + · · ·+ ‖DkF‖p,

Wt(ω) = ωt

〈h1, h2〉H :=m∑i=1

∫ ∞0

hi1(s)hi2(s)ds.

‖F‖k,p := ‖F‖p + ‖DF‖p + · · ·+ ‖DkF‖p,

Wt(ω) = ωt

〈h1, h2〉H :=m∑i=1

∫ ∞0

hi1(s)hi2(s)ds.

‖F‖k,p := ‖F‖p + ‖DF‖p + · · ·+ ‖DkF‖p,

Strong continuity

Let (Xλ)λ∈Λ be a family of Rd -valued Wiener functionals over W.Suppose that for some p > 1:(1) Xλ ∈ D2,p for each λ ∈ Λ, and λ 7→ ‖Xλ‖2,p is locally bounded.

(2) λ 7→ Xλ is continuous in probability, i.e., for any ε > 0 and λ0 ∈ Λ,

limλ→λ0

P(|Xλ − Xλ0 | ≥ ε) = 0.

(3) For each λ ∈ Λ, the Malliavin covariance matrix ΣXλ of Xλ is invertible

almost surely.Then the law of Xλ in Rd admits a density ρλ(x) so that λ 7→ ρλ iscontinuous in L1(Rd).

Strong continuity

Let (Xλ)λ∈Λ be a family of Rd -valued Wiener functionals over W.Suppose that for some p > 1:(1) Xλ ∈ D2,p for each λ ∈ Λ, and λ 7→ ‖Xλ‖2,p is locally bounded.(2) λ 7→ Xλ is continuous in probability, i.e., for any ε > 0 and λ0 ∈ Λ,

limλ→λ0

P(|Xλ − Xλ0 | ≥ ε) = 0.

Strong continuity

limλ→λ0

P(|Xλ − Xλ0 | ≥ ε) = 0.

almost surely.

Then the law of Xλ in Rd admits a density ρλ(x) so that λ 7→ ρλ iscontinuous in L1(Rd).

Strong continuity

limλ→λ0

P(|Xλ − Xλ0 | ≥ ε) = 0.

Remark

Bogachev (AMS, 2010) have already showed such a result in the first orderSobolev space W 1,p(Rd ,Rd) provided p ≥ d . Theorem 5.3 does notdepend on the dimension of space.

Thank you !

Derivative Formula, Coupling Property and Strong Feller ...pkim/7ICSAA/Slides/Zhao_Dong.pdf · Z....

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