Approximations of Stochastic Partial Differential Equations

Dept. of Mathematics University of Oslo

Pure Mathematics No 1

ISSN 0806–2439 February 2014

Approximations of Stochastic Partial Differential Equations

Giulia Di Nunno

Tusheng Zhang

February 7, 2014

Abstract

In this paper we show that solutions of stochastic partial differ- ential equations driven by Brownian motion can be approximated by stochastic partial differential equations forced by pure jump noise/random kicks. Applications to stochastic Burgers equations are discussed.

Key words: Stochastic partial differential equations; Approximations; Jump noise; Tightness; Weak convergence; Stochastic Burgers equations.

AMS Subject Classification: Primary 60H15 Secondary 93E20, 35R60.

1 Introduction

Stochastic evolution equations and stochastic partial differential equations (SPDEs) are of great interest to many people. There exists a great amount of literature on the subject, see, for example the monographs [PZ], [C].

In this paper, we consider the following stochastic evolution equation:

dYt = −AYtdt+ [b1(Yt) +b2(Yt)]dt+σ(Yt)dBt, (1.1)

Y₀ = h∈H, (1.2)

in the framework of a Gelfand triple :

V ⊂H ∼=H^∗ ⊂V^∗, (1.3)

where H, V are Hilbert spaces,A is the infinitesimal generator of a strongly continuous semigroup, b₁, σ are measurable mappings fromH intoH,b₂ is a measurable mappings fromH intoV^∗(the dual ofV),B_t, t≥0 is a Brownian motion. The solutions are considered to be weak solutions (in the PDE sense)

1 Department of Mathematics, University of Oslo, N-0316 Oslo, Norway

2 School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England, U.K. Email: tusheng.zhang@manchester.ac.uk

in the space V and not as mild solutions in H as is more common in the literature. The stochastic evolution equations of this type driven by Wiener processes were first studied by in [P] and subsequently in [KR]. For stochastic equations with general Hilbert space valued semimartingales replacing the Brownian motion we refer to [GK1], [GK2], [G] and [RZ].

The aim of this paper is to study the approximations of stochastic evolu- tion equations of the above type by solutions of stochastic evolution equations driven by pure jump processes, namely forced by random kicks. One of the motivations is to shine some light on numerical simulations of SPDEs driven by pure jump noise. To include interesting applications, the drift of the equa- tion (1.1) will consist of a “good” part b₁ and a “bad” part b₂. The crucial step of obtaining the approximation is to establish the tightness of the ap- proximating equations in the space of Hilbert space-valued right continuous paths with left limits. This is tricky because of the nature of the infinite dimensions and weak assumptions on the drift b₂. We first obtain the ap- proximations assuming the diffusion coefficient σ takes values in the smaller space V and then remove the restriction by another layer of approximations.

As far as we are aware of, this is the first paper to consider such approxi- mations for SPDEs. The approximations of small jump L´evy processes were considered in [AR]. Robustness of solutions of stochastic differential equa- tions replacing small jump Levy processes by Brownian motion was discussed in [BDK] and [DSE], and for the backward case in [DKV].

The rest of the paper is organized as follows. In Section 2 we lay down the precise framework. The main part is Section 3, where the approxima- tions are established and the applications to stochastic Burgers equations are discussed.

2 Framework

LetV,Hbe two separable Hilbert spaces such thatV is continuously, densely imbedded in H. IdentifyingH with its dual we have

V ⊂H ∼=H^∗ ⊂V^∗, (2.1)

whereV^∗ stands for the topological dual ofV. We assume that the imbedding V ⊂H is compact. Let A be a self-adjoint operator on the Hilbert space H satisfying the following coercivity hypothesis: There exist constants α₀ >0, α₁ >0 and λ₀ ≥0 such that

α0||u||²_V ≤2< Au, u >+λ0|u|²_H ≤α1||u||²_V for all u∈V . (2.2)

< Au, u >=Au(u) denotes the action of Au∈V^∗ on u∈V.

We remark that A is generally not bounded as an operator from H into H.

Let (Ω,F, P) be a probability space equipped with a filtration{F_t}satisfying

the usual conditions. Let {B_t, t≥0} be a real-valuedF_t- Brownian motion, ν(dx) a σ-finite measure on the measurable space (R₀,B(R₀)), where R₀ = R\ {0}. Let p= (p(t)), t ∈ Dp be a stationary Ft-Poisson point process on R₀ with characteristic measure ν. Here D_p represents a countable (random) subset of (0,∞). See [IW] for the details on Poisson point processes. Denote byN(dt, dx) the Poisson counting measure associated with p, i.e.,N(t, A) = P

s∈Dp,s≤tI_A(p(s)). Let ˜N(dt, dx) :=N(dt, dx)−dtν(dx) be the compensated Poisson random measure. Let b₁, σ be measurable mappings from H into H, and b2(·) a measurable mapping fromV into V^∗. Denote byD([0, T], H) the space of all c`adl`ag paths from [0, T] into H equipped with the Skorohod topology. Consider the stochastic evolution equation:

dX_t = −AX_tdt+ [b₁(X_t) +b₂(X_t)]dt+σ(X_t)dB_t, (2.3)

X₀ = h∈H. (2.4)

Introduce the following conditions:

(H.1) There exists a constant C <∞ such that

|b₁(y₁)−b₁(y₂)|²_H +|σ(y₁)−σ(y₂)|²_H

≤ C|y₁−y₂|²_H, for all y₁, y₂ ∈H. (2.5) (H.2) b₂(·) is a mapping fromV into V^∗ that satisfies

(i)< b2(u), u >= 0 for u∈V,

(ii) There exist constants C₁,β < ¹₂ such that

< b₂(y₁)−b₂(y₂), y₁−y₂ >

≤ βα₀||y₁−y₂||²_V +C₁|y₁−y₂|²_H(||y₁||²_V +||y₂||²_V)

for all y1, y2 ∈V, (2.6)

(iii) There exists a constant 0< γ < 1 such that||b2(u)||V^∗ ≤C2|u|^2−γ_H ||u||^γ_V for u∈V.

Under the assumptions (H.1) and (H.2), it is known that equations (2.3) admits a unique solution.

We finish this section with two examples.

Example 2.1 LetDbe a bounded domain inR^d. SetH =L²(D). LetV = H₀^1,2(D) denote the Sobolev space of order one with homogenous boundary conditions. Denote by a(x) = (a_ij(x)) a symmetric matrix-valued function on D satisfying the uniform ellipticity condition:

1

cId ≤a(x)≤cId for some constant c∈(0,∞).

Define

Au=−div(a(x)∇u(x)).

Then (2.2) is fulfilled for (H, V, A).

Example 2.2 Let Au = −∆_α, where ∆_α denotes the generator of a sym- metricα-stable process inR^d, 0< α≤2. ∆_α is called the fractional Laplace operator. It is well known that the Dirichlet form associated with ∆α is given by

E(u, v) = K(d, α) Z Z

R^d×R^d

(u(x)−u(y))(v(x)−v(y))

|x−y|^d+α dxdy, D(E) ={u∈L²(R^d) :

Z Z

R^d×R^d

|u(x)−u(y)|²

|x−y|^d+α dxdy <∞},

where K(d, α) = α2^α−3π^−d+22 sin(^απ₂ )Γ(^d+α₂ )Γ(^α₂). To study equation (2.3), we choose H = L²(R^d), and V = D(E) with the inner product < u, v >=

E(u, v) + (u, v)_L²_(R^d₎. Then (2.2) is fulfilled for (H, V, A). See [FOT] for details about the fractional Laplace operator.

3 Approximations of SPDEs by pure jump type SPDEs

Set, for ε∈(0,1),

α() = Z

{|x|≤}

x²ν(dx) ¹₂

Consider the following SPDE driven by pure jump noise:

X_t^ε = h− Z t

0

AX_s^εds+ Z t

0

[b₁(X_s^ε) +b₂(X_s^ε)]ds + 1

α() Z t

0

Z

|x|≤ε

σ(X_s−^ε )xN˜(ds, dx). (3.7) Under the assumptions (H.1) and (H.2), the SPDE above admits a unique solution. See [RZ], [LR] and also [AWZ]. Let X denote the solution to the SPDE (2.3):

X_t = h− Z t

0

AX_sds+ Z t

0

[b₁(X_s) +b₂(X_s)]ds+ Z t

0

σ(X_s)dB_s. (3.8) Denote byµ_ε,µrespectively the laws ofX^ε andXon the spacesD([0, T], H) and C([0, T], H). Consider the following conditions:

(H.3) There exists a sequence of mappings σ_n(·) :H →V such that

(i)|σn(y1)−σn(y2)|H ≤c|y1−y2|H, where cis a constant independent of n,

(ii) |σ_n(y)−σ(y)|_H →0 uniformly on bounded subsets ofH.

Remark 3.1 In most of the cases, one simply chooses σ_n to be the finite dimensional projection of σ into the space V.

(H.3)⁰ The mappingσ(·) takes the spaceV into itself and satisfies||σ(y)||_V ≤ c(1 +||y||_V) for some constant c.

(H.4) There exists an orthonormal basis {e_k, k ≥1} of H such that Ae_k = λ_ke_k and 0≤λ₁ ≤λ₂ ≤...≤λ_n→ ∞ as n→ ∞.

We first prepare some preliminary results needed for the proofs of the main theorems.

The following estimate holds for {X^ε, ε >0}.

Lemma 3.2 Let X^ε be the solution of equation (3.7). If _α()^ε ≤C₀ for some constant C0, then we have for p≥2,

sup

ε

{E[ sup

0≤t≤T

|X_t^ε|^p_H] +E[

Z T 0

||X_s^ε||²_Vds

p 2

]}<∞. (3.9) Proof. We prove the lemma for p = 4. Other cases are similar. In view of the assumption (H.2), by Ito’s formula, we have

|X_t^ε|²_H

= |h|²_H −2 Z t

0

< X_s^ε, AX_s^ε > ds+ 2 Z t

0

< X_s^ε, b₁(X_s^ε)> ds +

Z t 0

Z

|x|≤ε

| 1

α()σ(X_s−^ε )x|²_H + 2 < X_s−^ε , 1

α()σ(X_s−^ε )x >

N˜(ds, dx) +

Z t 0

Z

|x|≤ε

| 1

α()σ(X_s−^ε )x|²_Hdsν(dx). (3.10)

Let Mt=

Z t 0

Z

|x|≤ε

| 1

α()σ(X_s−^ε )x|²_H + 2< X_s−^ε , 1

α()σ(X_s−^ε )x >

N˜(ds, dx).

By Burkh¨older’s inequality, fort ≤T, and a positive constant C, we have E[ sup

0≤u≤t

|M_u|²_H]≤CE[[M, M]_t]

= CE[

Z t 0

Z

|x|≤ε

| 1

α()σ(X_s−^ε )x|²_H + 2< X_s−^ε , 1

α()σ(X_s−^ε )x >

2

N(ds, dx)]

= CE[

Z t 0

Z

|x|≤ε

| 1

α()σ(X_s−^ε )x|²_H + 2< X_s−^ε , 1

α()σ(X_s−^ε )x >

2

dsν(dx)]

≤ CE[

Z t 0

(1 +|X_s^ε|⁴_H)ds], (3.11)

where the linear growth condition on σ and the fact _α()^ε ≤ C₀ have been used. Use first (2.2) and then square both sides of the resulting inequality

to obtain from (3.10) that

|X_t^ε|⁴_H + Z t

0

||X_s^ε||²_Vds ²

≤ C_T|h|⁴_H +C_T Z t

0

(1 +|X_s^ε|⁴_H)ds+C_TM_t². (3.12) Take superemum over the interval [0, t] in (3.12), use (3.11) to get

E[ sup

0≤s≤t

|X_s^ε|⁴_H] +E[

Z t 0

||X_s^ε||²_Vds ²

]

≤ C|h|⁴_H +CE[

Z t 0

(1 +|X_s^ε|⁴_H)ds]. (3.13) Applying Gronwall’s inequality proves the lemma.

Proposition 3.3 Assume (H.1), (H.2), (H.3)⁰,(H.4) and _α()^ε ≤C0 for some constant C₀. Then the family {X^ε, ε >0} is tight on the space D([0, T], H).

Proof. Write

Y_t^ε = 1 α()

Z t 0

Z

|x|≤ε

σ(X_s−^ε )xN˜(ds, dx), (3.14) and set Z_t^ε = X_t^ε −Y_t^ε. It suffices to prove that both {Y^ε, ε > 0} and {Z^ε, ε >0}are tight. This is done in two steps.

Step 1. Prove that {Y^ε, ε >0} is tight.

In view of the assumptions onσ (H.3)⁰, we haveY^ε ∈D([0, T], V). Since the imbedding V ⊂ H is compact, according to Theorem 3.1 in [J], it is sufficient to show that for every e ∈ H, {< Y^ε, e >, ε > 0} is tight in D([0, T], R). Note that

sup

ε

E[ sup

0≤t≤T

< Y_t^ε, e >²]≤sup

ε

E[ sup

0≤t≤T

|Y_t^ε|²_H]

≤ Csup

ε

1 α()²E[

Z T 0

Z

|x|≤ε

|σ(X_s^ε)|²_Hx²ν(dx)ds]

= Csup

ε

E[

Z T 0

|σ(X_s^ε)|²_Hds]<∞, (3.15) and for any stoping times τ_ε≤T and any positive constantsδ_ε →0 we have

E[|< Y_τ^ε_ε, e >−< Y_τ^ε_ε+δε, e >|²]≤ 1 α()²E[

Z τε+δε

τε

Z

|x|≤ε

|σ(X_s^ε)|²_Hx²ν(dx)ds]

≤ Cδ_εsup

ε

E[(1 + sup

0≤t≤T

|X_t^ε|²_H)]→0, (3.16)

as ε → 0. By Theorem 3.1 in [J], (3.15) and (3.16) yields the tightness of

< Y^ε, e >, ε >0.

Step 2. Prove that {Z^ε, ε >0}is tight.

It is easy to see that Z^ε satisfies the equation:

Z_t^ε = h− Z t

0

AZ_s^εds− Z t

0

AY_s^εds +

Z t 0

b₁(Z_s^ε+Y_s^ε)ds+ Z t

0

b₂(Z_s^ε+Y_s^ε)ds. (3.17) Recall {e_k, k ≥1} is the othonormal basis of H consisting of eigenvectors of A (see (H.4)). We have

< Z_t^ε, ek >

= < h, e_k>−λ_k Z t

0

< Z_s^ε, e_k> ds−λ_k Z t

0

< Y_s^ε, e_k> ds +

Z t 0

< b₁(Z_s^ε+Y_s^ε), e_k> ds+ Z t

0

< b₂(Z_s^ε+Y_s^ε), e_k> ds.(3.18) By Corollary 5.2 in [J], to obtain the tightness of {Z^ε, ε > 0} we need to show

(i). {< Z^ε, e_k >, ε >0} is tight in D([0, T], R) for every k, (ii). for any δ >0,

N→∞lim sup

ε

P( sup

0≤t≤T

R^ε_N(t)> δ) = 0, (3.19) where

R^ε_N(t) =

∞

X

k=N

< Z_t^ε, e_k >² .

The proof of (i) is similar to that of the tightness of < Y^ε, e >, ε > 0. It is omitted. Let us prove (ii). By the chain rule, it follows that

< Z_t^ε, e_k >² = < h, e_k>² −2λ_k Z t

0

< Z_s^ε, e_k >² ds−2λ_k Z t

0

< Y_s^ε, e_k >< Z_s^ε, e_k > ds +2

Z t 0

< b₁(Z_s^ε+Y_s^ε), e_k>< Z_s^ε, e_k > ds +2

Z t 0

< b₂(Z_s^ε+Y_s^ε), e_k>< Z_s^ε, e_k > ds. (3.20) By the variation of constants formula, we have

< Z_t^ε, e_k >² = e^−2λkt < h, e_k>² −2λ_k Z t

0

e^{−2λk(t−s)}< Y_s^ε, e_k>< Z_s^ε, e_k> ds +2

Z t 0

e^{−2λk(t−s)} < b₁(Z_s^ε+Y_s^ε), e_k >< Z_s^ε, e_k > ds +2

Z t 0

e^{−2λk(t−s)} < b₂(Z_s^ε+Y_s^ε), e_k >< Z_s^ε, e_k > ds. (3.21)

Hence

R_N^ε(t) =

∞

X

k=N

< Z_t^ε, e_k>²

=

∞

X

k=N

e^−2λkt< h, e_k >² −2 Z t

0

∞

X

k=N

λ_ke^{−2λk(t−s)} < Y_s^ε, e_k >< Z_s^ε, e_k > ds

+2 Z t

0

∞

X

k=N

e^{−2λk(t−s)}< b1(Z_s^ε+Y_s^ε), ek>< Z_s^ε, ek > ds +2

Z t 0

∞

X

k=N

e^{−2λk(t−s)}< b₂(Z_s^ε+Y_s^ε), e_k>< Z_s^ε, e_k > ds

=: I_N⁽¹⁾(t) +I_N⁽²⁾(t) +I_N⁽³⁾(t) +I_N⁽⁴⁾(t). (3.22) Obviously

I_N⁽¹⁾(t)≤

∞

X

k=N

< h, e_k>²→0, (3.23) as N → ∞. For the third term on the right side of (3.22), we have

|I_N⁽³⁾(t)| ≤ 2 Z t

0

e^{−2λN(t−s)}

∞

X

k=N

|< b₁(Z_s^ε+Y_s^ε), e_k>< Z_s^ε, e_k >|ds

≤ 2 Z t

0

e^{−2λN(t−s)}ds( sup

0≤s≤T

|Z_s^ε|_H)( sup

0≤s≤T

|b₁(Z_s^ε+Y_s^ε)|_H)

≤ C 1 λ_N

1 + sup

0≤s≤T

|Z_s^ε|²_H + sup

0≤s≤T

|Y_s^ε|²_H

. (3.24)

Hence,

sup

ε

E[ sup

0≤t≤T

|I_N⁽³⁾(t)|]

≤ C 1 λN

1 + sup

ε

E[ sup

0≤s≤T

|Z_s^ε|²_H] + sup

ε

E[ sup

0≤s≤T

|Y_s^ε|²_H]

→0, as N → ∞. (3.25)

Let us turn to I_N⁽²⁾(t). By H¨older’s inequality,

|I_N⁽²⁾(t)| ≤ 2 Z t

0

(

∞

X

k=N

e^{−4λk(t−s)}λ_k< Z_s^ε, e_k >²)¹²

×(

∞

X

k=N

λ_k < Y_s^ε, e_k >²)¹²ds

≤ 2 Z t

0

(

∞

X

k=N

e^{−4λk(t−s)}λ_k< Z_s^ε, e_k >²)¹²(< AY_s^ε, Y_s^ε >²)¹²ds

≤ C( sup

0≤s≤T

||Y_s^ε||_V) Z t

0

e^{−λN(t−s)}(

∞

X

k=N

e^{−2λk(t−s)}λ_k < Z_s^ε, e_k >²)¹²ds

≤ C( sup

0≤s≤T

||Y_s^ε||_V)( sup

0≤s≤T

|Z_s^ε|_H) Z t

0

e^{−λN(t−s)} 1

√t−sds

≤ C( 1

√λN

Z ∞ 0

e^−u 1

√udu)( sup

0≤s≤T

||Y_s^ε||_V)( sup

0≤s≤T

|Z_s^ε|_H). (3.26) In view of the assumption (H.2), the last term on the right side of (3.22) can be estimated as follows:

|I_N⁽⁴⁾(t)| = | Z t

0

∞

X

k=N

e^{−2λk(t−s)}< b₂(X_s^ε), e_k>< Z_s^ε, e_k> ds|

= | Z t

0

∞

X

k=N

e^{−2λk(t−s)}p

λ₀+λ_k <(A+λ₀I)⁻¹²b₂(X_s^ε), e_k >< Z_s^ε, e_k > ds|

≤ C Z t

0

∞

X

k=N

e^{−4λk(t−s)}<(A+λ₀I)⁻¹²b₂(X_s^ε), e_k >²

!¹₂

×

∞

X

k=N

(λ₀+λ_k)< Z_s^ε, e_k >²

!¹₂ ds

≤ C Z t

0

||Z_s^ε||_Ve^{−2λN(t−s)}

∞

X

k=N

<(A+λ₀I)⁻¹²b₂(X_s^ε), e_k>²

!¹₂ ds

≤ C Z t

0

||Z_s^ε||_Ve^{−2λN(t−s)}||b₂(X_s^ε)||_V^∗ds

≤ C Z t

0

||Z_s^ε||_Ve^{−2λN(t−s)}|X_s^ε|^2−γ_H ||X_s^ε||^γ_Vds. (3.27)

This yields that

|I_N⁽⁴⁾(t)| ≤ C sup

0≤s≤T

|X_s^ε|^2−γ_H Z t

0

||Z_s^ε||_Ve^{−2λN(t−s)}||X_s^ε||^γ_Vds

≤ C sup

0≤s≤T

|X_s^ε|^2−γ_H Z t

0

e^{−2λN(t−s)}(||X_s^ε||^1+γ_V +||X_s^ε||^γ_V||Y_s^ε||_V)ds

≤ C sup

0≤s≤T

|X_s^ε|^2−γ_H Z t

0

e^{−1−γ4 λN(t−s)}ds ^1−γ₂

× Z T

0

||X_s^ε||^1+γ_V +||X_s^ε||^γ_V||Y_s^ε||V

_1+γ² ds

1+γ 2

≤ C( 1

λ_N)^1−γ2 sup

0≤s≤T

|X_s^ε|^2−γ_H Z T

0

||X_s^ε||^1+γ_V +||X_s^ε||^γ_V||Y_s^ε||_V1+γ² ds

^1+γ₂

(3.28) Hence,

sup

ε

E[ sup

0≤t≤T

|I_N⁽⁴⁾(t)|]

≤ C( 1

λ_N)^1−γ2 sup

ε

E

sup

0≤s≤T

|X_s^ε|^2−γ_H

× Z T

0

||X_s^ε||^1+γ_V +||X_s^ε||^γ_V||Y_s^ε||_V1+γ² ds

1+γ 2





≤ C( 1

λ_N)^1−γ2 sup

ε

E

sup

0≤s≤T

|X_s^ε|^2−γ_H

× Z T

0

C||X_s^ε||²_V +c||Y_s^ε||²_V ds

1+γ 2





→0, as N → ∞, (3.29)

where we used the fact that

|ab| ≤C(|a|^p +|b|^q), 1 p+ 1

q = 1.

Putting together (3.22)—(3.29) and applying the Chebychev inequality we obtain (3.19).

Let D denote the class of functions f ∈ C_b³(H) that satisfy (i) f⁰(z) ∈ D(A) and |Af⁰(z)|_H ≤C(1 +|z|_H) for some constantC, where f⁰(z) stands for the Frechet derivative of f, (ii) f⁰⁰, f⁰⁰⁰ are bounded.

Forf ∈ D, define

L^εf(z) =−< Af⁰(z), z > +< b₁(z), f⁰(z)>+< b₂(z), f⁰(z)>

+ Z

|x|≤ε

f(z+ 1

α()σ(z)x)−f(z)−< f⁰(z), 1

α()σ(z)x >

ν(dx), (3.30) and

Lf(z) = −< Af⁰(z), z >+< b₁(z), f⁰(z)>+< b₂(z), f⁰(z)>

+1

2 < f⁰⁰(z)σ(z), σ(z)> . (3.31) Lemma 3.4 Assume lim_{ε→0 α(ε)}^ε = 0. For f ∈ D, it holds that

L^εf(z)→Lf(z) uniformly on bounded subsets of H (3.32) as ε→0.

Proof. Note that

f(y+w)−f(y)−< f⁰(y), w >=

Z 1 0

dα Z α

0

< f⁰⁰(y+βw)w, w > dβ.

Thus

L^εf(z)−Lf(z)

= Z

|x|≤ε

Z 1 0

dα Z α

0

dβ < f⁰⁰(z+β 1

α()σ(z)x) 1

α()σ(z)x, 1

α()σ(z)x > ν(dx)

− Z 1

0

dα Z α

0

dβ < f⁰⁰(z)σ(z), σ(z)>

= 1

α()² Z

|x|≤ε

x²ν(dx) Z 1

0

dα Z α

0

dβ

< f⁰⁰(z+β 1

α()σ(z)x)σ(z), σ(z)>

−< f⁰⁰(z)σ(z), σ(z)>] (3.33)

Hence, for z ∈B_N ={z ∈H;|z|_H ≤N}we have

|L^εf(z)−Lf(z)|

≤ C 1 α()²

Z

|x|≤ε

x²ν(dx) Z 1

0

dα Z α

0

dββ 1

α()|σ(z)|_H|x||σ(z)|²_H

≤ C_N ε

α() →0, (3.34)

uniformly onBN asε→0, where we have used the local Lipschtiz continuity of f⁰⁰(z).

Theorem 3.5 Suppose (H.1), (H.2),(H.3)⁰, (H.4) hold and limε→0 ε α(ε) = 0.

Then, for any T > 0, µ_ε converges weakly to µ, for ε → 0, on the space D([0, T], H) equipped with the Skorohod topology.

Proof. Since the mapping σtakes values in the spaceV, by Proposition 3.3, the family {µ_ε, ε > 0} is tight. Let µ₀ be the weak limit of any convergent sequence {µ_εn} on the canonical space (Ω = D([0, T], H),F) as ε_n → 0.

We will show that µ₀ = µ. Denote by X_t(ω) = w(t), ω ∈ Ω the coordinate process. Set J(X) = sup_0≤s≤T|X_s−Xs−|_H. Since

E^µε[J(X)] = E[J(X^ε)]

≤ ε

α()E[ sup

0≤s≤T

|σ(X_s^ε)|_H]

≤ C ε

α()(1 +E[ sup

0≤s≤T

|X_s^ε|_H])→0, (3.35) as ε → 0, it follows from Theorem 13.4 in [B] that µ0 is supported on the C([0, T], H), the space of H-valued continuous functions on [0, T]. As a consequence, the finite dimensional distributions of µ_εn converge to that of µ0.

Letf ∈ D. By Ito’s formula, f(X_t^ε)−f(h)−

Z t 0

L^εf(X_s^ε)ds

= Z t

0

Z

|x|≤ε

{f(X_s−^ε + 1

α()σ(X_s−^ε )x)−f(X_s−^ε )}N˜(ds, dx) (3.36) is a martingale. Hence, for any s₀ < s₁ < ... < s_n ≤s < t and f₀, f₁, ...f_n ∈ C_b(H) it holds that

E^µε[

f(X_t)−f(X_s)− Z t

s

L^εf(X_u)du

f(X_s0)...f(X_sn)]

= 0. (3.37)

For any positive constant M >0, by Lemma 3.4 we have

n→∞lim E^µεn[ Z t

s

|L^εnf(X_u)−Lf(X_u)|du, sup

0≤u≤T

|X_u|_H ≤M] = 0. (3.38) On the other hand, in view of the assumptions on f we have

sup

n

E^µεn[ Z t

s

|L^εnf(X_u)−Lf(X_u)|du, sup

0≤u≤T

|X_u|_H > M]

≤ C 1 M sup

n

E^µεn[ sup

0≤u≤T

|X_u|³_H]≤C⁰ 1

M (3.39)

Combining (3.38) with (3.39) we arrive at

n→∞lim E^µεn[ Z t

s

|L^εnf(X_u)−Lf(X_u)|du] = 0. (3.40) By the weak convergence of µ_εn and the convergence of finite distributions, it follows from (3.37) and (3.40) that

E^µ0[(f(X_t)−f(X_s)− Z t

s

Lf(X_u)du)f(X_s0)...f(X_sn)]

= lim

n→∞E^µεn[(f(X_t)−f(X_s)− Z t

s

Lf(X_u)du)f(X_s0)...f(X_sn)]

= lim

n→∞E^µεn[(f(X_t)−f(X_s)− Z t

s

L^εnf(X_u)du)f(X_s0)...f(X_sn)]

= 0. (3.41)

Since s₀ < s₁ < ... < s_n ≤ s < t are arbitrary, (3.41) implies that for any f ∈ D,

M_t^f =f(X_t)−f(h)− Z t

0

Lf(X_s)ds, t ≥0,

is a martingale under µ₀. In particular, let f(z) =< e_k, z > and f(z) =<

e_k, z >< e_j, z > respectively to obtain that under µ₀ M_t^k := < e_k, X_t>−< e_k, h >+

Z t 0

< Ae_k, X_s > ds− Z t

0

< b₁(X_s), e_k > ds

− Z t

0

< b₂(X_s), e_k> ds (3.42)

and

M_t^k,j :=< e_k, X_t>< e_j, X_t>−< e_k, h >< e_j, h >

+ Z t

0

{< Ae_k, X_s>< e_j, X_s>+< Ae_j, X_s>< e_k, X_s>}ds

− Z t

0

< b₁(X_s), e_k < e_j, X_s>+e_j < e_k, X_s >> ds

− Z t

0

< b₂(X_s), e_k < e_j, X_s>+e_j < e_k, X_s >> ds

− Z t

0

< σ(Xs), ek >< σ(Xs), ej > ds (3.43) are martingales. This together with Ito’s formula yields that

< M^k, M^j >t= Z t

0

< σ(Xs), ek >< σ(Xs), ej > ds, (3.44)

where< M^k, M^j >stands for the sharp bracket of the two martingales. Now by Theorem 18.12 in [K] (or Theorem 7.1⁰ in [IW]), there exists a probability space (Ω⁰,F⁰, P⁰) with a filtration F_t⁰ such that on the standard extension

(Ω×Ω⁰,F × F⁰,F_t× F_t⁰, µ₀×P⁰)

of (Ω,F,F_t, P) there exists a Brownian motionB_t, t≥0 such that M_t^k=

Z t 0

< σ(X_s), e_k > dB_s, (3.45) namely,

< e_k, X_t >−< e_k, h >

= −

Z t 0

< Ae_k, X_s> ds+ Z t

0

< b₁(X_s), e_k > ds+ Z t

0

< b₂(X_s), e_k> ds +

Z t 0

< σ(X_s), e_k > dB_s (3.46)

for any k ≥ 1. Thus, under µ₀, X_t, t ≥ 0 is a weak solution (both in the probabilistic and in PDE sense) of the SPDE:

X_t=h− Z t

0

AX_sds+ Z t

0

b₁(X_s)ds+ Z t

0

b₂(X_s)ds+ Z t

0

σ(X_s)dB_s By the uniqueness of the above equation, we conclude thatµ₀ =µcompleting the proof of the theorem.

Theorem 3.6 Suppose (H.1), (H.2), (H.3) and (H.4) hold andlimε→0 ε α(ε) = 0. Then, for any T > 0, µ_ε converges weakly to µ, for ε → 0, on the space D([0, T], H) equipped with the Skorohod topology.

Proof. Let σ_n(·) be the mapping specified in (H.3). Let X^n,ε, Xⁿ be the solutions of the SPDEs:

X_t^n,ε = h− Z t

0

AX_s^n,εds+ Z t

0

b₁(X_s^n,ε)ds+ Z t

0

b₂(X_s^n,ε)ds + 1

α() Z t

0

Z

|x|≤ε

σ_n(X_s−^n,ε)xN˜(ds, dx). (3.47)

X_tⁿ = h− Z t

0

AX_sⁿds+ Z t

0

b₁(X_sⁿ)ds+ Z t

0

b₂(X_sⁿ)ds +

Z t 0

σ_n(X_sⁿ)dB_s. (3.48)

We claim that for any δ >0,

n→∞lim sup

ε

P( sup

0≤t≤T

|X_t^n,ε−X_t^ε|> δ) = 0. (3.49)

n→∞lim P( sup

0≤t≤T

|X_tⁿ−X_t|² > δ) = 0. (3.50) Let us only prove (3.49). The proof of (3.50) is simpler. As the proof of (3.9), we can show that

sup

n

sup

ε

{E[ sup

0≤t≤T

|X_t^n,ε|²_H] +E[

Z T 0

||X_s^n,ε||²_Vds]}<∞. (3.51)

sup

n

{E[ sup

0≤t≤T

|X_tⁿ|²_H] +E[

Z T 0

||X_sⁿ||²_Vds]}<∞. (3.52) By Ito’s formula, we have

e^{−γR0t(||Xsn,ε||2V+||Xsε||2V)ds}|X_t^n,ε−X_t^ε|²_H

= −γ Z t

0

e^{−γR0s(||Xun,ε||2V+||Xuε||2V)du}|X_s^n,ε−X_s^ε|²_H(||X_s^n,ε||²_V +||X_s^ε||²_V)ds

−2 Z t

0

e^{−γR0s(||Xun,ε||2V+||Xuε||2V)du}< X_s^n,ε−X_s^ε, A(X_s^n,ε−X_s^ε)> ds +2

Z t 0

e^{−γR0s(||Xun,ε||2V+||Xuε||2V)du} < X_s^n,ε−X_s^ε, b₁(X_s^n,ε)−b₁(X_s^ε)> ds +2

Z t 0

e^{−γR0s(||Xun,ε||2V+||Xuε||2V)du} < X_s^n,ε−X_s^ε, b₂(X_s^n,ε)−b₂(X_s^ε)> ds +

Z t 0

Z

|x|≤ε

e^{−γR0s(||Xun,ε||2V+||Xuε||2V)du}

| 1

α()(σ_n(X_s−^n,ε)x−σ(X_s−^ε )x)|²_H +2<(X_s^n,ε−X_s^ε), 1

α()(σn(X_s−^n,ε)x−σ(X_s−^ε )x)>

N˜(ds, dx) +

Z t 0

Z

|x|≤ε

e^{−γR0s(||Xun,ε||2V+||Xuε||2V)du}| 1

α()(σn(X_s−^n,ε)x−σ(X_s−^ε )x)|²_Hdsν(dx) :=

6

X

k=1

I_k^n,ε(t). (3.53)

In view of the assumption (2.6), we see that I₁^n,ε(t) +I₂^n,ε(t) +I₄^n,ε(t)

≤ −(1−2β)α0

Z t 0

e^−γ

R_s

0(||Xu^n,ε||²_V+||X_u^ε||²_V)du||X_s^n,ε−X_s^ε||²_Vds, (3.54) if γ ≥2C₁, whereC₁ is the constant appeared in (2.6).