LijunBo KehuaShi andYongjinWang SupporttheoremforastochasticCahn-Hilliardequation ∗

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El e c t ro nic

Journ a l of

Pr

ob a b il i t y

Vol. 15 (2010), Paper no. 17, pages 484–525.

Journal URL

http://www.math.washington.edu/~ejpecp/

Support theorem for a stochastic Cahn-Hilliard equation

^∗

Lijun Bo¹ Kehua Shi^2,†and Yongjin Wang³

1Department of Mathematics, Xidian University, Xi’an 710071, China bolijunnk@yahoo.com.cn

2School of Mathematical Sciences, Xiamen University, Xiamen 361005, China kehuashink@gmail.com

3School of Mathematical Sciences, Nankai University, Tianjin 300071, China yjwang@nankai.edu.cn

Abstract

In this paper, we establish a Stroock-Varadhan support theorem for the global mild solution to a d (d≤3)-dimensional stochastic Cahn-Hilliard partial differential equation driven by a space- time white noise.

Key words: Stochastic Cahn-Hilliard equation, Space-time white noise, Stroock-Varadhan support theorem.

AMS 2000 Subject Classification:Primary 60H15, 60H05.

Submitted to EJP on November 7, 2009, final version accepted Aril 15, 2010.

∗The research of K. Shi and Y. Wang was supported by the LPMC at Nankai University and the NSF of China (No.

10871103). The research of L. Bo was supported by the Fundamental Research Fund for the Central Universities (No.

JY10000970002).

†Corresponding author. Email: kehuashink@gmail.com

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1 Introduction and main result

In this paper, we consider the following stochastic Cahn-Hilliard equation:







∂u/∂t=−∆

∆u+f(u)

+σ(u)W˙, in[0,T]×D, u(0) =ψ,

∂u/∂n=∂[∆u]/∂n=0, on[0,T]×∂D,

(1.1)

where ∆ denotes the Laplace operator, the domain D= [0,π]^d (d = 1, 2, 3), and f :R→Ris a polynomial of degree 3 with positive dominant coefficient (which is due to the background of the equation from material science). Assume thatσ:R→Ris a bounded and Lipschitzian function and W˙ is a Gaussian space-time white noise on some complete probability space(Ω,F,P)satisfying

E

W˙(x,t)W˙(y,s)

=δ(|t−s|)δ(|x−y|), (t,x),(s,y)∈[0,T]×D.

Hereδ(·)is the Dirac delta function concentrated at the point zero.

The (deterministic) Cahn-Hilliard equation (i.e.,σ≡0 in (1.1)) has been extensively studied (see, e.g.,[2; 3; 4; 5; 10; 15; 18]) as a well-known model of the macro-phase separation that occurs in an isothermal binary fluid, when a spatially uniform mixture is quenched below a critical temperature at which it becomes unstable. A stochastic version of the Cahn-Hilliard equation (when σ ≡ 1 in (1.1)) was first proposed by Da Prato and Debussche [8], and the existence, uniqueness and regularity of the global mild solution were explored. In Cardon-Weber[6], the authors considered this type of stochastic equation in a general case onσ, which is equivalent to the following form:

u(t,x) = Z

D

G_t(x,y)ψ(y)dy+ Z t

0

Z

D

∆G_t−s(x,y)f(u(s,y))dyds +

Z t

0

Z

D

G_t₋_s(x,y)σ(u(s,y))W(dy, ds), (1.2) where G_t(·,∗) denotes the Green kernel corresponding to the operator∂ /∂t+ ∆² with the homo- geneous Neumann’s boundary condition as in (1.1). Since Stroock and Varadhan[17]established their famous support theorem for diffusion processes, there have been many research works on this issue, for example, a variety of support theorems for 1-dimensional second-order parabolic and hyperbolic stochastic partial differential equations (abbr. SPDEs) have been discussed in the literature (see, e.g.,[1; 7; 13; 14]). Millet and Sanz-Solé [13] characterized the support of the law of the solution to a class of hyperbolic SPDEs, which simplified the proof in[17]. In Bally et al. [1], the authors proved a support theorem for a semi-linear parabolic SPDE. Moreover, a support result for a generalized Burgers SPDE (containing a quadratic term) was established in Cardon-Weber and Millet[7]. Herein, we are attempting to establish a support theorem of the law corresponding to the solution to Equation (1.1) inC([0,T],L^p([D]))forp≥4. The main strategy used in this paper is an approximation procedure by using a space-time polygonal interpolation for the white noise, and we particularly adopt a localization argument, which was used in [7] for studying a support theorem of a Burgers-type equation. However here we need more technical estimates concerning the high-order Green kernelG_t(·,∗), which sharp the estimates in[6](see Appendix).

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In what follows, we introduce the main result of this paper. To do it, we define the following Cameron-Martin spaceH by

H =

¨

h(t,x) = Z t

0

Z

Qd i=1[0,x_i]

h(s,y)dyds; (t,x) = (t,(x₁, . . . ,x_d))∈[0,T]×D,

h∈L²([0,T]×D)

« , and the corresponding norm by

khk_H = s

Z T

0

Z

D

|h(s,y)|²dyds, for all h∈ H.

LetHb represent the subset of H, in which the first-order derivative h ofh∈ H is bounded. For h∈ H, consider the following skeleton equation:

S(h)(t,x) = Z

D

G_t(x,y)ψ(y)dy+ Z t

0

Z

D

∆G_t−s(x,y)f(S(h)(s,y))dyds +

Z t

0

Z

D

G_t₋_s(x,y)σ(S(h)(s,y))h(s,y)dyds. (1.3) Recall Equations (1.1) and (1.2). We make the following assumptions throughout the paper:

(H1). Assume thatσ:R→Ris bounded and belongs to C³(R)with bounded first to third-order partial derivatives, and

(H2). The initial functionψ∈L^p(D)forp≥4, andψis%∈]0, 1]–H¨older continuous.

Now we are at the position to state the main result of this paper.

Theorem 1.1. Under the assumptions (H1) and (H2), let u = (u(t,x))_(t,x_)∈[0,T]×D be the unique solution to Equation(1.2)in C([0,T],L^p(D))with p≥4and P◦u⁻¹ denote the law (a probability measure)of the solution u. Recall the skeleton equation(1.3), and setS_H ={S(h);h∈ H }. Then we have

(a) Let p>6. Then forα¯∈]0, min{¹₂(1−^d₄),^%

4}[andβ¯∈]0, min{2−^d₂,%}[, the topological support supp(P◦u⁻¹)in C^α^¯^{, ¯}^β([0,T]×D)of the lawP◦u⁻¹is the closure ofS_H.

(b) Let p ≥ 4. Then for α¯ ∈]0, min{¹₂(1− ^d₄),^%

4}[, the topological support supp(P◦u⁻¹) in C^α^¯([0,T],L^p(D))of the lawP◦u⁻¹is the closure ofS_H.

The rest of this paper is organized as follows: In the coming section, we give a difference approximation to the(d+1)–dimensional space-time white noise ˙W(x,t)and study some concrete properties of the approximating noises. In Section 3, we introduce a localization framework as in [7], and then switch to prove the support theorem by checking the conditions (C1) and (C2) below (see Section 3). Sections 4 and 5 are devoted to checking the validity of the conditions (C1) and (C2), respectively. In Section 6, we prove the continuity of the solutionS(h)to the skeleton equation (1.3) inHband finally we complete the proof of Theorem 1.1.

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2 Difference approximation to white noise

In this section, we give a difference approximation to the (d+1)-dimensional space-time white noise W˙, which is a space-time polygonal interpolation for ˙W.

Letn∈Nandt∈[0,T], set t_n= max

j∈{0,1,...,2ⁿ}

¦j T2⁻ⁿ;j T2⁻ⁿ≤t©

, and t_n=

t_n−T2⁻ⁿ

∨0.

Let k := (k₁, . . . ,k_d) ∈ I^d_n := {0, 1, . . . ,n−1}^d. Define a partition (4j,k)j=0,1,...,2ⁿ⁻¹,k∈I^d_n of OT := [0,T]×Dby

4j,k=D_k×]j T2⁻ⁿ,(j+1)T2⁻ⁿ], where D_k = Q_d

j=1]kjπn⁻¹,(kj+1)πn⁻¹]. For x_j ∈]kjπn⁻¹,(kj+1)πn⁻¹]with j = 1, . . . ,d, we set D_k(x) =Qd

j=1]k_jπn⁻¹,(k_j +1)πn⁻¹]. Further, for each (t,x) ∈ OT, we define the following difference approximation to ˙W by

W˙_n(x,t) =

( _W₍₄

j−1,k)

|⁴j−1,k| , (x,t)∈ 4j,k, j=1, . . . , 2ⁿ−1, k∈I^d_n, 0, (x,t)∈ 40,k, k∈I^d_n,

(2.1)

where 4j,k

= Tπ^d(n^d2ⁿ)⁻¹ is the volume of the partition4j,k for each j = 0, 1, . . . , 2ⁿ−1 and k∈I^d_n.

Next we suppose that

(H3). the mappings F,H,K : R → R are bounded, globally Lipschitzian and H ∈ C³(R) with bounded first to third-order derivatives.

We now consider the following equations forh∈ Hb, X_n(t,x) = G_t∗ψ(x)

+ Z t

0

Z

D

G_t₋_s(x,y)

F(Xn(s,y))W(dy, ds) +H(Xn(s,y))Wn(dy, ds) +

Z t

0

Z

D

G_t−s(x,y)h

K(X_n(s,y))h(s,y)

−H˙(X_n(s,y))[αn(s,y)F(X_n(s,y)) +βn(s,y)H(X_n(s,y))]i dyds +

Z t

0

Z

D

∆_yG_t−s(x,y)f(X_n(s,y))dyds, (2.2) and

X(t,x) = G_t∗ψ(x) + Z t

0

Z

D

G_t₋_s(x,y)[F+H](X(s,y))W(dy, ds) +

Z t

0

Z

D

G_t−s(x,y)K(X(s,y))h(s,y)dyds

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+ Z t

0

Z

D

∆_yG_t−s(x,y)f(X(s,y))dyds, (2.3)

where

G_t∗ψ(x):= Z

D

G_t(x,y)ψ(y)dy, and for eachn∈N,

αn(t,x) := n^d2ⁿ(Tπ^d)⁻¹ Z t_n

tn

Z

Dk(x)

G_t−s(x,y)dyds, β_n(t,x) := n^d2ⁿ(Tπ^d)⁻¹

Z t

t_n

Z

Dk(x)

G_t−s(x,y)dyds.

Forαn(t,x)andβn(t,x), by virtue of (A.4) in Lemma A.1, we claim that sup

(t,x)∈OT

|αn(t,x)| ≤ C n^d, and (2.4)

sup

(t,x)∈OT

|βn(t,x)| ≤C n^d. (2.5)

Indeed, using (A.4), we have for each t∈[0,T], sup

x∈D|αn(t,x)| ≤ C n^d2ⁿmax

k∈I^d_n

( 1_D_k(x)

Z t_n

t_n

Z

Dk

|G_t−s(x,y)|dyds )

≤ C n^d2ⁿ|t_n−t_n|

≤ C n^d, and

sup

x∈D|βn(t,x)| ≤C n^d2ⁿ|t−t_n| ≤C n^d, follows from the equality (A.19).

In the following, letF= (Ft)0≤t≤T be the natural filtration generated byW, i.e., Ft=σ{W(B×[0,s]);s∈[0,t], B∈ B(D)}.

Then for every t ∈ [0,T] and n ∈ N fixed, (W˙_n(x,t))x∈D given by (2.1) is Ft-adapted. More precisely, it isFt_n-adapted and which is independent of the informationFtn.

Lemma 2.1. For each fixed n∈Nand p≥1, we have sup

(t,x)∈OT

E

|W˙_n(x,t)|^p

≤C_pn^{d p}²2^np².

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Proof. By virtue of the definition (2.1), sup

(t,x)∈OT

E

|W˙_n(x,t)|^p

= sup

(t,x)∈OT

E

2ⁿ−1

X

j=1

X

k∈I^d_n

W(4_j−1,k)

|4j−1,k| 1₄_j

−1,k(x,t)

p

≤ C_pmax





 E

W(4_j−1,k)

|4j−1,k|

p

; j=1, . . . , 2ⁿ−1, k∈I^d_n





 .

Note that for each j=1, . . . , 2ⁿ−1 andk∈I^d_n, W(4j−1,k)

|4_j−1,k| ∼N(0,|4j−1,k|⁻¹). For any random variableZ∼N(0,σ²), it holds that

E|Z|^p= 1 pπ

p

2^pσ^2pΓ p

2+1 2

, whereΓdenotes the Gamma function. This yields that

sup

(t,x)∈OT

E

|W˙_n(x,t)|^p

≤C_pmaxnp

|4_j−1,k|^−p; j=1, . . . , 2ⁿ−1, k∈I^d_no ,

and which proves the lemma.

Letn∈Nbe fixed. Forα >0 and t∈]0,T], we now define an event ¯Ω^α_n,t by Ω¯^α_n,t=

¨

ω∈Ω; sup

(s,y)∈[0,t]×D

W˙(y,s;ω)

≤αn^d2ⁿ²

«

. (2.6)

For this event, we have Lemma 2.2. If chooseα >2

qlog 2 Tπ^d, then

nlim→∞Ph

Ω¯^α_n,Tic

=0.

Proof. Let Z ∼ N(0, 1) be a standard normal random variable. Then according to the definition (2.1) for ˙W_n(x,t),

Ph

Ω¯^α_n,Tic

= P

(j,k)∈{1,...,2maxⁿ−1}×I^d_n

¨W(4j−1,k)

|4_j−1,k|

«

≥αn^d2²ⁿ

≤ n^d2ⁿP

|Z| ≥αp

Tπ^dn^d/²

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= n^d2ⁿP

|Z|²

4 ≥ α²Tπ^d 4 n^d

≤ n^d2ⁿexp

−α²Tπ^d 4 n^d

E

exp

|Z|² 4

. (2.7)

Note thatE h

exp |Z|²

4

i=p

2. Then (2.7) further yields that 0≤Ph

Ω¯^α_n,Tic

≤ p

2n^dexp

nlog 2− α²Tπ^d 4 n^d

≤ p

2n^dexp

n^d

log 2−α²Tπ^d 4

→ 0, as n→ ∞, (2.8)

ifα >2 qlog 2

Tπ^d. Thus the proof of the lemma is complete.

3 Localization framework

In this section, we adopt a localization method used in[7]to deal with Equation (1.1). In addition, we will prove a key proposition, which is useful in the proof of Theorem 1.1.

Proposition 3.1. Under the assumptions (H1) and(H2), let X = (X(t,x))₍t,x)∈[0,T]×D (resp. Xn) be the unique solution to Equation(2.3)(resp.(2.2))in C([0,T],L^p(D))with p≥4. Recall the skeleton equation(1.3), and setS_H ={S(h);h∈ H }. Then we have

(i) Let p > 6. Then for α¯ ∈]0, min{¹₂(1− ^d₄),^%₄}[ and β¯ ∈]0, min{2− ^d₂,%}[, the sequence X_n converges in probability to X in C^α^¯^{, ¯}^β([0,T]×D).

(ii) Let p≥4. Then forα¯∈]0, min{¹₂(1− ^d₄),^%₄}[, the sequence Xn converges in probability to X in C^α^¯([0,T],L^p(D)).

Next we give a sketch for the proof of the conclusion(ii)in Proposition 3.1. The similar argument can also be used to prove the part(i). For(t,x)∈ OT, set

Y_n(t,x):=X_n(t,x)−X(t,x). From (2.2) and (2.3), it follows that

Y_n(t,x) =

3

X

i=1

Γⁱ_n(t,x) + Λ_n(t,x), (3.1)

where

Γ¹_n(t,x) := Z t

0

Z

D

G_t−s(x,y)

(F+H)(X_n(s,y))−(F+H)(X(s,y))

W(dy, ds), Γ²_n(t,x) :=

Z t

0

Z

D

G_t₋_s(x,y)

K(Xn(s,y))−K(X(s,y))

h(s,y)dyds,

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Γ³_n(t,x) := Z t

0

Z

D

∆_yG_t−s(x,y)

f(X_n(s,y))− f(X(s,y)) dyds, and

Λn(t,x) := Z t

0

Z

D

G_t−s(x,y)H(X_n(s,y))

W_n(dy, ds)−W(dy, ds)

− Z t

0

Z

D

G_t₋_s(x,y)H˙(Xn(s,y))

×

αn(s,y)F(X_n(s,y)) +βn(s,y)H(X_n(s,y))

dyds. (3.2)

Introduce an auxiliaryFtn-adapted process

X_n⁻(t,x):=G_t₋_t_n x,X_n(tn,·)

, for (t,x)∈ OT. (3.3)

Recall the localization argument adopted in[7]. Forγ∈(0, 1)andp≥4, define Φ^p,_n^γ(t):= sup

s∈[0,t]kY_n(t,·)kp+ sup

s6=s⁰∈[0,t]

kY_n(s,·)−Y_n(s⁰,·)k_p

|s−s⁰|^γ , wherek · kp corresponds to the norm ofL^p(D)and forδ >0,

τ^δ_n:=inf¦

t>0; Φ^p,_n^γ(t)≥δ©

∧T.

ForM> δ, define the following events A_t(M−δ) :=

¨

ω∈Ω; sup

s∈[0,t]kX(s,·)kp≤M−δ

«

, (3.4)

A^M_n(t) :=

¨

ω∈Ω; sup

s∈[0,t]kX_n(s,·)kp∨ sup

s∈[0,t]kX(s,·)kp≤M

«

. (3.5)

Then fort∈]0,T],

A_t(M−δ)∩ {t≤τ^δ_n} ⊆A^M_n(t). (3.6) In fact, from the inequality|y| ≤ |x−y|+|x|, it follows that

A_t(M−δ)∩ {t≤τ^δ_n}

⊆

¨ sup

s∈[0,t]kX(s,·)kp≤M−δ

«

∩

¨ sup

s∈[0,t]kX_n(s,·)−X(s,·)kp≤δ

«

⊆

¨ sup

s∈[0,t]kX(s,·)kp≤M

«

∩

¨ sup

s∈[0,t]kX_n(s,·)kp≤M

«

= A^M_n(t).

Recall the event ¯Ω^α_n,t defined by (2.6) in Section 2 and thatα >2 qlog 2

Tπ^d. For each fixedδ >0 and V ∈C^γ([0,T];L^p(D)), set

kVk_γ,p,τ^δ_n := sup

s∈[0,T∧τ^δ_n]kV(s,·)kp+ sup

s6=s⁰∈[0,T∧τ^δ_n]

kV(s,·)−V(s⁰,·)kp

|s−s⁰|^γ .

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Then forq≥1, P

Φ^p,_n^γ(T)≥δ

≤ P

Φ^p,_n^γ(T)≥δ, A_T(M−δ)∩Ω¯^α_n,T +P

A^c_T(M−δ) +P

hΩ¯^α_n,Tic

= P

kY_nk_γ,p,τ^δ_n≥δ, A_T(M−δ)∩Ω¯^α_n,T +P

A^c_T(M−δ)

+Ph

Ω¯^α_n,Tic

≤ δ^−qE

1_A

T(M−δ)∩Ω¯^αn,TkY_nk^q_γ_,p,_τδ n

+P

A^c_T(M−δ)

+Ph

Ω¯^α_n,Tic , However, Lemma 2.2 and Lemma 3.1 (the latter will be proved below) yield that

P

A^c_T(M−δ)

+Ph

Ω¯^α_n,Tic

→0, asM → ∞, n→ ∞. Therefore, by Lemma A.1 in[7]and (3.6), in order to prove

Φ^p,_n^γ(T)→0, in probability, asn→ ∞, (3.7) it suffices to check that there existq≥ pandθ >qα¯(we have setγ=α¯, where ¯αis the exponent presented in Proposition 3.1) such that

(C1) ∀t∈[0,T], lim

n→∞Eh

1A¯^M_n(t)kY_n(t,·)k^q_pi

=0;

(C2) ∀s<t∈[0,T], Eh

1A¯^M_n(t)kY_n(t,·)−Y_n(s,·)k^q_pi

≤C|t−s|^1+θ. Here the event ¯A^M_n(t)is defined by

A¯^M_n (t) =A^M_n(t)∩Ω¯^α_n,t, t∈[0,T], (3.8) and which satisfies the order relation:

A¯^M_n (t)⊂A¯^M_n(r), ifr≤t.

Lemma 3.1. Let the event A_T(M)be defined by(3.4). Then

M→∞lim P

A_T(M)c

=0.

Proof. Note that forp≥4,β∈[p,∞[andβ∈]p, 6p/(6−p)⁺[(ifd=3), P

A_T(M)c

=P

sup

t∈[0,T]kX(t,·)kp≥M

≤M^−βE

sup

t∈[0,T]kX(t,·)k^β_p

. Therefore, it remains to prove

E

sup

t∈[0,T]kX(t,·)k^β_p

<∞. (3.9)

Define for(t,x)∈ OT,

L₁(u)(t,x) := Z t

0

Z

D

G_t₋_s(x,y)[F+H](u(s,y))W(dy, ds);

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L₂(u)(t,x) := Z t

0

Z

D

G_t−s(x,y)K(u(s,y))h(s,y)dyds, and setZ=X−L(X)withL=L₁+L₂. Then







∂Z

∂t + ∆²Z−∆f([Z+L(X)]) =0, Z(0) =ψ,

∂Z/∂n=∂[∆Z]/∂n=0, on∂D.

(3.10)

Further, the Garsia-Rodemich-Rumsey lemma (see, e.g., Theorem B.1.1 and Theorem B.1.5 in[9]) yields that, if for anyq,δ∈]1,∞[and someγ⁰,γ⁰⁰∈]0, 1],

(a) sup

(t,x)∈OT

E

|L(u)(t,x)|^2qδ

<∞, (b) E

|L(u)(t,x)−L(u)(t⁰,x⁰)|^2q

≤C

|t−t⁰|^γ⁰⁰+|x−x⁰|^γ⁰q

, q>1,

then (3.9) holds. So we only need to prove (a) and (b). Note thatKis bounded andd≤3. Then in light of (A.4),

sup

(t,x)∈OT

Eh

L₂(u)(t,x) ^2qδ

i

≤ kKk^2qδ_∞ sup

(t,x)∈OT

Z t

0

Z

D

|G_t₋_s(x,y)h(s,y)|dyds

2qδ

≤ kKk^2qδ_∞ khk^qδ_HT⁽¹⁻^d⁴⁾^q^δ<∞.

If sett>t⁰, then by (A.7)–(A.9) in Lemma A.2, we have forγ⁰∈[0, 1−d/4[andγ⁰⁰∈[0, 2∧(4−d)[, E

|L₂(u)(t,x)−L₂(u)(t⁰,x⁰)|^2q

≤ C

|t−t⁰|^γ⁰⁰+|x−x⁰|^γ⁰q

.

The estimate ofL₁is similar to that of L₂ (or see[6]). Thus the proof the lemma is complete.

4 Auxiliary lemmas

In this section, we present a sequence of auxiliary lemmas for checking the conditions (C1) and (C2) under (H1)–(H3) given in Section 1 and 2. Throughout Sections 4–6, (H1)–(H3) are assumed to be satisfied.

The following lemma tells us that, to check(C1), it suffices to show(C1)holds withΛn(t,x)instead ofY_n(t,x).

Lemma 4.1. Assume p≥4, and q≥ p if d=1, 2, and q∈]p, 6p/(6−p)⁺[if d =3. Then for each n∈N,

sup

t∈[0,T]

Eh

1A¯^M_n(t)kY_n(t,·)k^q_pi

≤C sup

t∈[0,T]

Eh

1A¯^M_n(t)kΛn(t,·)k^q_pi

, (4.1)

whereΛn(t,x)is defined by(3.2).

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Proof. Note that for eacht∈[0,T], Eh

kY_n(t,·)k^q_pi

≤CE



 X3

i=1

1A¯^M_n(t)kΓⁱ_n(t,·)k^q_p+1A¯^M_n(t)kΛ_n(t,·)k^q_p



,

whereΓⁱ(t,x)(i=1, 2, 3) are defined in (3.1). Therefore by (A.16) in Lemma A.3, for ¹

r = ²_q−²_p+ 1∈[0, 1],

E h

1A¯^M_n(t)kΓ¹_n(t,·)k^q_pi

≤ C Z

D

Eh

1A¯^M_n(t)|Γ¹_n(t,x)|^qi dx

≤ CE







Z t

0

Z

D

G²_t₋_s(·,y)1A¯^M_n(s)|Y_n(s,y)|²dyds

q 2







≤ CE

Z t

0

(t−s)^4r^d⁻^d²1A¯^M_n(s)kY_n(s,·)k²_pds

q 2

≤ C

Z t

0

(t−s)^4r^d⁻^d²ds

q−2 2 Z t

0

(t−s)^4r^d⁻^d² sup

r∈[0,s]

Eh

1A¯^M_n(r)kY_n(r,·)k^q_pi ds

. Similarly, we have

Eh

1A¯^M_n(t)kΓ²_n(t,·)k^q_pi

≤ Ckhk^q_HE







Z t

0

Z

D

G²_t₋_s(·,y)1A¯^M_n(s)|Y_n(s,y)|²dyds

q 2







≤ C

Z t

0

(t−s)^4r^d⁻^d²ds

^q−

2 2 Z t

0

(t−s)^4r^d⁻^d² sup

r∈[0,s]

Eh

. As forΓ³_n(t,x), using (A.13) with ¹

r2 =¹_q− ¹_ρ+1∈[0, 1], Eh

1A¯^M_n(t)kΓ³_n(t,·)k^q_pi

≤ CE





1A¯^M_n(t)

Z t

0

Z

D

∆_yG(t−s,·,y)[f(X_n(s,·))−f(X(s,·))]dyds

q







≤ CE



1A¯^M_n(t)

Z t

0

(t−s)^4r^d²⁻^d+2⁴ kf(Xn(s,·))−f(X(s,·))k_ρds

q

. (4.2)

Note thatu(s,·),v(s,·)∈L^p(D), for eachs∈[0,T], we have forρ= ₃^p, ku(s,·)−v(s,·)k_ρ ≤ Cku(s,·)−v(s,·)kp,

ku²(s,·)−v²(s,·)k_ρ ≤ Cku(s,·)−v(s,·)k_p

ku(s,·)k_p+kv(s,·)k_p

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ku³(s,·)−v³(s,·)k_ρ ≤ Cku(s,·)−v(s,·)kp

×h

ku(s,·)k²_p+kv(s,·)k²_p+ku(s,·)kpkv(s,·)kp

i . Hence from (4.2), it follows that

Eh

1A¯^M_n(t)kΓ³_n(t,·)k^q_pi

≤ CE

Z t

0

(t−s)^4r^d²⁻^d+2⁴ 1A¯^M_n(s)kY_n(s,·)k_pds

q

≤ C

Z t

0

(t−s)^4r^d²⁻^d+2⁴ ds

q−1Z t

0

(t−s)^4r^d²⁻^d+2⁴ sup

r∈[0,s]

Eh

. Note that the following equivalent relations holds:

d 4r − d

2+1>0 ⇔ 1 q > 1

p −1 6, d

4r₂− d+2

4 +1>0 ⇔ 1 q > 1

p +1 2− 2

d.

Then the desired result follows from the Gronwall’s lemma.

Recall theFt_n-adapted processX⁻_n(t,x)defined by (3.3). Then we have Lemma 4.2. Let q≥p≥6. Then there exists a constant C:=C_M>0such that

sup

(t,x)∈OT

Eh

1A¯^M_n(t)|X_n(t,x)−X_n⁻(t,x)|^qi

≤C2^−nqι, (4.3)

whereι:= ¹₂(1− ^d₄).

Proof. Recall (2.2) and (3.3), and we get

X_n(t,x)−X⁻_n(t,x) =

4

X

k=1

T_n^k(t,x), where

T_n¹(t,x) := Z t

t_n

Z

D

G_t₋_s(x,y)F(Xn(s,y))W(dy, ds),

T_n²(t,x) := Z t

t_n

Z

D

G_t₋_s(x,y)H(X_n(s,y))W_n(dy, ds),

T_n³(t,x) := Z t

t_n

Z

D

∆yG_t−s(x,y)f(X_n(s,y))dyds,

T_n⁴(t,x) := Z t

t_n

Z

D

G_t−s(x,y)K_n(s,y)dyds,

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and

K_n(s,y) = K(X_n(s,y))h(s,y)

−(FH˙)(X_n(s,y))α_n(s,y)−(HH)(X˙ _n(s,y))β_n(s,y). From (2.4) and (2.5), it follows that forh∈ Hb,

sup

(s,y)∈OT

|K_n(s,y)| ≤C n^d. (4.4)

On the other hand, using (A.4) and the boundedness ofF, E

|T_n¹(t,x)|^q

= E





Z t

tn

Z

D

G_t−s(x,y)F(X_n(s,y))W(dy, ds)

q



≤ CE







Z t

t_n

Z

D

G²_t₋_s(x,y)F²(X_n(s,y))dyds

q/2





≤ C



 Z t

t_n

Z

D

G²_t₋_s(x,y)dyds





q/2

≤ C



 Z t

t_n

(t−s)⁻^d⁴ds





q/2

≤ C2⁻¹²^nq(¹⁻^d⁴⁾. (4.5)

Further, the Hölder inequality, Lemma 2.1 and the boundedness ofH jointly imply that E

|T_n²(t,x)|^q

= E





Z t

t_n

Z

D

G_t₋_s(x,y)H(Xn(s,y))Wn(dy, ds)

q



≤ CE





Z t

t_n

Z

D

|G_t−s(x,y)||W˙_n(y,s)|dyds

q



≤ CE









 Z t

t_n

Z

D

|G_t₋_s(x,y)||W˙_n(y,s)|^qdyds



·



 Z t

t_n

Z

D

|G_t₋_s(x,y)|dyds





q−1





≤ C



 Z t

t_n

Z

D

|G_t−s(x,y)|dyds





q

sup

(t,x)∈OT

E

|W˙_n(x,t)|^q

≤ C n^dq²2^nq²|t−t_n|^q

≤ C n

dq

22⁻²^nq. (4.6)

Note thatf is a polynomial of degree 3. Then by virtue of (A.13) withκ= _∞¹ −³_p+1=1−³_p ∈[0, 1], Eh

1A¯^M_n(t)|T_n³(t,x)|^qi

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= E



1A¯^M_n(t)

Z t

tn

Z

D

∆_yG_t−s(x,y)f(X_n(s,y))dyds

q



≤ E







 Z t

t_n

(t−s)^d⁴^κ−^d+2⁴ 1A¯^M_n(s)kf(Xn(s,·))k^p

3ds





q



≤ E







 Z t

tn

(t−s)^d⁴⁽¹⁻³^p⁾⁻^d+2⁴ 1A¯^M_n(s)

h

kX_n(s,·)k_p+kX_n(s,·)k²_p+kX_n(s,·)k³_pi ds





q



≤ C



 Z t

t_n

(t−s)⁻⁽¹²⁺^3d^4p⁾ds





q

≤ C2^−nq(

1

2−^3d_4p). (4.7)

Thanks to (4.4), we conclude that E

|T_n⁴(t,x)|^q

= E





Z t

t_n

Z

D

G_t₋_s(x,y)Kn(s,y)dyds

q



≤ C n^dq



 Z t

t_n

Z

D

|G_t−s(x,y)|dyds





q

≤ C n^dq|t−t_n|^q

≤ C n^dq2^−nq. (4.8)

Thus the estimate (4.3) follows from (4.5)–(4.8).

Remark 4.1. We can easily check that there exists a constant C_M >0such that for q≥p≥4, sup

t∈[0,T]

Eh

1A¯^M_n(t)kX_n(t,·)−X_n⁻(t,·)k^q_pi

≤C_M2^−nqι, (4.9)

whereι= ¹₂(1−^d₄). In fact, for the proof of(4.9), the only estimate that needs to be checked is that of T_n⁴. Apply(A.13)withκ= ¹_p− ³_p+1=1−²_p ∈[0, 1]to T_n⁴ and we can get(4.9).

Next we prove a useful lemma, which will be used frequently later. For(t,x),(s,y)∈ OT, set η(t,s,x,y)

(·,∗) := 1_[_0,t](·)G_t−·(x,∗)−1_[_0,s](·)G_s−·(y,∗), (4.10)

η(e t,s,x,y)

(·,∗) := 1_[_0,t](·)∆G_t−·(x,∗)−1_[_0,s](·)∆G_s−·(y,∗). (4.11) LetV be anF= (Ft)0≤t≤T-predictable process. For 0≤s≤t≤T and x,y ∈D, define

λ^k_n(V)(t,s,x,y):=

Z ^(k+1)T

2n ∧t

kT 2n∧t

Z

D

η(t,s,x,y)

(r,z)V(r,z)W_n(dz, dr), (4.12)

withk=0, 1, . . . , 2ⁿ−1. Then we have