**E**l e c t ro nic

**J**ourn a l
of

**P**r

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 Bo^{1} Kehua Shi^{2,†}and Yongjin Wang^{3}

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 sup-
port 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

**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**→

**R**is a polynomial of degree 3 with positive dominant coefficient (which is due to the background of the equation from material science). Assume that

*σ*:

**R**→

**R**is 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.*

Here*δ(·)*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 on*σ, which is equivalent to the following form:*

*u*(*t,x*) =
Z

*D*

*G** _{t}*(

*x*,

*y*)ψ(

*y*)d

*y*+ Z

*t*

0

Z

*D*

∆*G** _{t−s}*(

*x*,

*y*)

*f*(

*u*(

*s,y*))d

*y*ds +

Z *t*

0

Z

*D*

*G*_{t}_{−}* _{s}*(x,

*y*)σ(u(s,

*y*))W(d

*y, ds)*, (1.2) where

*G*

*(·,∗) denotes the Green kernel corresponding to the operator*

_{t}*∂ /∂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 hy- perbolic 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) in

*C*([0,

*T]*,

*L*

*([D]))for*

^{p}*p*≥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 kernel

*G*

*(·,∗), which sharp the estimates in[6](see Appendix).*

_{t}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

Q*d*
*i=*1[0,x* _{i}*]

**h**(*s,y*)d*y*ds; (*t*,*x*) = (*t,*(*x*_{1}, . . . ,*x** _{d}*))∈[0,

*T*]×

*D,*

**h**∈*L*^{2}([0,*T*]×*D)*

« , and the corresponding norm by

k*h*k_{H} =
s

Z *T*

0

Z

*D*

|**h**(s,*y*)|^{2}d*y*ds, for all *h*∈ H.

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

*S*(*h*)(*t,x*) =
Z

*D*

*G** _{t}*(

*x*,

*y*)ψ(

*y*)d

*y*+ 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*)d

*yds.*(1.3) Recall Equations (1.1) and (1.2). We make the following assumptions throughout the paper:

(H1). Assume that*σ*:**R**→**R**is bounded and belongs to *C*^{3}(**R**)with bounded first to third-order
partial derivatives, and

(H2). The initial function*ψ*∈*L** ^{p}*(

*D*)for

*p*≥4, and

*ψ*is

*%*∈]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*≥4

*and*

**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 for*α*¯∈]0, min{^{1}_{2}(1−^{d}_{4}),^{%}

4}[*andβ*¯∈]0, min{2−^{d}_{2},*%}[, the topological support*
supp(**P**◦*u*^{−}^{1})*in C*^{α}^{¯}^{, ¯}* ^{β}*([0,

*T*]×

*D)of the law*

**P**◦

*u*

^{−}

^{1}

*is the closure of*S

_{H}

*.*

**(b)** *Let p* ≥ 4. *Then for* *α*¯ ∈]0, min{^{1}_{2}(1− ^{d}_{4}),^{%}

4}[, the topological support supp(**P**◦*u*^{−}^{1}) *in*
*C*^{α}^{¯}([0,*T*],*L** ^{p}*(D))

*of the law*

**P**◦

*u*

^{−}

^{1}

*is the closure of*S

_{H}

*.*

The rest of this paper is organized as follows: In the coming section, we give a difference approxima-
tion 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 solution*S(h)*to the skeleton equation (1.3)
inHband finally we complete the proof of Theorem 1.1.

**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*.

Let*n*∈**N**and*t*∈[0,*T*], set
*t** _{n}*= max

*j∈{*0,1,...,2* ^{n}*}

¦*j T*2^{−}* ^{n}*;

*j T*2

^{−}

*≤*

^{n}*t*©

, and *t** _{n}*=

*t** _{n}*−

*T*2

^{−}

**

^{n}∨0.

Let **k** := (*k*_{1}, . . . ,*k** _{d}*) ∈

**I**

^{d}*:= {0, 1, . . . ,*

_{n}*n*−1}

*. Define a partition (4*

^{d}*j,k*)

*j*=0,1,...,2

^{n−}^{1},k∈

**I**

^{d}*of O*

_{n}*T*:= [0,

*T*]×

*D*by

4*j,k*=*D*** _{k}**×]

*j T*2

^{−n},(

*j*+1)

*T2*

^{−n}], where

*D*

**= Q**

_{k}

_{d}*j=1*]k*j**πn*^{−}^{1},(k*j*+1)πn^{−}^{1}]. For *x** _{j}* ∈]k

*j*

*πn*

^{−}

^{1},(k

*j*+1)πn

^{−}

^{1}]with

*j*= 1, . . . ,

*d, we*set

*D*

**(**

_{k}*x*) =Q

*d*

*j=*1]*k*_{j}*πn*^{−1},(*k** _{j}* +1)π

*n*

^{−1}]. Further, for each (

*t*,

*x*) ∈ O

*T*, we define the following difference approximation to ˙

*W*by

*W*˙* _{n}*(

*x*,

*t*) =

( _{W}_{(4}

*j−*1,k)

|^{4}*j−*1,k| , (*x*,*t*)∈ 4*j,k*, *j*=1, . . . , 2* ^{n}*−1,

**k**∈

**I**

^{d}*, 0, (x,*

_{n}*t)*∈ 40,k,

**k**∈

**I**

^{d}*,*

_{n}(2.1)

where
4*j,k*

= *Tπ** ^{d}*(n

*2*

^{d}*)*

^{n}^{−}

^{1}is the volume of the partition4

*j,k*for each

*j*= 0, 1, . . . , 2

*−1 and*

^{n}**k**∈

**I**

^{d}*.*

_{n}Next we suppose that

(H3). the mappings *F,H,K* : **R** → **R** are bounded, globally Lipschitzian and *H* ∈ *C*^{3}(R) with
bounded first to third-order derivatives.

We now consider the following equations for*h*∈ Hb,
*X** _{n}*(t,

*x*) =

*G*

*∗*

_{t}*ψ(x*)

+
Z *t*

0

Z

*D*

*G*_{t}_{−}* _{s}*(

*x*,

*y*)

*F*(X*n*(s,*y*))W(d*y, ds) +H(X**n*(s,*y))W**n*(d*y, 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*))[α

*n*(

*s,y*)

*F*(

*X*

*(*

_{n}*s,y*)) +

*β*

*n*(

*s,y*)

*H*(

*X*

*(*

_{n}*s,y*))]i d

*yds*+

Z *t*

0

Z

*D*

∆_{y}*G** _{t−s}*(

*x,y*)

*f*(X

*(s,*

_{n}*y*))d

*yds,*(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

+
Z *t*

0

Z

*D*

∆_{y}*G** _{t−s}*(x,

*y*)

*f*(X(s,

*y*))d

*yds,*(2.3)

where

*G** _{t}*∗

*ψ(x*):= Z

*D*

*G** _{t}*(x,

*y*)ψ(

*y*)d

*y,*and for each

*n*∈

**N,**

*α**n*(*t*,*x*) := *n** ^{d}*2

*(*

^{n}*Tπ*

*)*

^{d}^{−}

^{1}Z

*t*

_{n}*t**n*

Z

*D***k**(x)

*G** _{t−s}*(

*x*,

*y*)dyds,

*β*

*(t,*

_{n}*x*) :=

*n*

*2*

^{d}*(T*

^{n}*π*

*)*

^{d}^{−}

^{1}

Z *t*

*t*_{n}

Z

*D***k**(x)

*G** _{t−s}*(x,

*y*)d

*yds.*

For*α**n*(t,*x*)and*β**n*(t,*x*), by virtue of (A.4) in Lemma A.1, we claim that
sup

(*t,x*)∈O*T*

|α*n*(*t,x*)| ≤ *C n** ^{d}*, and (2.4)

sup

(t,x)∈O*T*

|β*n*(t,*x*)| ≤*C n** ^{d}*. (2.5)

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

*x*∈*D*|α*n*(*t*,*x*)| ≤ *C n** ^{d}*2

*max*

^{n}**k**∈**I**^{d}_{n}

(
**1**_{D}** _{k}**(

*x*)

Z *t*_{n}

*t*_{n}

Z

*D***k**

|*G** _{t−s}*(

*x*,

*y*)|d

*yds*)

≤ *C n** ^{d}*2

*|*

^{n}*t*

*−*

_{n}*t*

*|*

_{n}≤ *C n** ^{d}*,
and

sup

*x*∈*D*|β*n*(*t,x*)| ≤*C n** ^{d}*2

*|*

^{n}*t*−

*t*

*| ≤*

_{n}*C n*

*, follows from the equality (A.19).*

^{d}In the following, letF= (F*t*)0≤*t*≤*T* be the natural filtration generated by*W*, i.e.,
F*t*=*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 F

*t*-adapted. More precisely, it isF

*t*

*-adapted and which is independent of the informationF*

_{n}*t*

*n*.

**Lemma 2.1.** *For each fixed n*∈**N***and p*≥1, we have
sup

(t,x)∈O*T*

**E**

|*W*˙* _{n}*(x,

*t*)|

**

^{p}≤*C*_{p}*n*^{d p}^{2}2^{np}^{2}.

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

(t,x)∈O*T*

**E**

|*W*˙* _{n}*(x,

*t*)|

**

^{p}= sup

(t,x)∈O*T*

**E**

2* ^{n}*−1

X

*j=1*

X

**k**∈**I**^{d}_{n}

*W*(4* _{j−1,k}*)

|4*j*−1,k| **1**_{4}_{j}

−1,k(*x*,*t*)

*p*

≤ *C** _{p}*max

**E**

*W*(4* _{j−}*1,k)

|4*j−1,k*|

*p*

; *j*=1, . . . , 2* ^{n}*−1,

**k**∈

**I**

^{d}

_{n}

.

Note that for each *j*=1, . . . , 2* ^{n}*−1 and

**k**∈

**I**

^{d}*,*

_{n}*W*(4

*j*−1,k)

|4* _{j−}*1,k| ∼

*N*(0,|4

*j*−1,k|

^{−}

^{1}). For any random variable

*Z*∼

*N*(0,

*σ*

^{2}), it holds that

**E**|*Z*|* ^{p}*= 1
p

*π*

p

2^{p}*σ*^{2p}Γ
*p*

2+1 2

, whereΓdenotes the Gamma function. This yields that

sup

(*t,x*)∈O*T*

**E**

|*W*˙* _{n}*(

*x*,

*t*)|

**

^{p}≤*C** _{p}*maxnp

|4* _{j−}*1,k|

^{−p};

*j*=1, . . . , 2

*−1,*

^{n}**k**∈

**I**

^{d}*o ,*

_{n}and which proves the lemma.

Let*n*∈**N**be fixed. For*α >*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** ^{d}*2

^{n}^{2}

«

. (2.6)

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

qlog 2
*T**π*^{d}*, then*

*n*lim→∞**P**h

Ω¯^{α}* _{n,T}*i

*c*

=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*),

**P**h

Ω¯^{α}* _{n,T}*i

*c*

= **P**

(j,k)∈{1,...,2max* ^{n}*−1}×

**I**

^{d}

_{n}¨*W*(4*j*−1,k)

|4* _{j−}*1,k|

«

≥*αn** ^{d}*2

^{2}

^{n}

≤ *n** ^{d}*2

^{n}**P**

|*Z*| ≥*α*p

*Tπ*^{d}*n*^{d/}^{2}

= *n** ^{d}*2

^{n}**P**

|*Z*|^{2}

4 ≥ *α*^{2}*Tπ** ^{d}*
4

*n*

^{d}

≤ *n** ^{d}*2

*exp*

^{n}

−*α*^{2}*Tπ** ^{d}*
4

*n*

^{d}
**E**

exp

|*Z*|^{2}
4

. (2.7)

Note that**E**
h

exp
|Z|^{2}

4

i=p

2. Then (2.7) further yields that
0≤**P**h

Ω¯^{α}* _{n,T}*i

*c*

≤ p

2n* ^{d}*exp

*nlog 2*− *α*^{2}*Tπ** ^{d}*
4

*n*

^{d}

≤ p

2n* ^{d}*exp

*n*^{d}

log 2−*α*^{2}*Tπ** ^{d}*
4

→ 0, as *n*→ ∞, (2.8)

if*α >*2
qlog 2

*T**π** ^{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. X*n*) *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{^{1}_{2}(1− ^{d}_{4}),^{%}_{4}}[ *and* *β*¯ ∈]0, min{2− ^{d}_{2},*%}[, the sequence X*_{n}*converges in probability to X in C*^{α}^{¯}^{, ¯}* ^{β}*([0,

*T*]×

*D).*

**(ii)** *Let p*≥4. Then for*α*¯∈]0, min{^{1}_{2}(1− ^{d}_{4}),^{%}_{4}}[, the sequence X*n* *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*)∈ O*T*, 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

Γ^{i}* _{n}*(t,

*x*) + Λ

*(t,*

_{n}*x*), (3.1)

where

Γ^{1}* _{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*(d*y, ds*),
Γ^{2}* _{n}*(t,

*x*) :=

Z *t*

0

Z

*D*

*G*_{t}_{−}* _{s}*(x,

*y*)

*K(X**n*(s,*y))*−*K(X*(s,*y*))

**h**(s,*y)*dyds,

Γ^{3}* _{n}*(t,

*x*) := Z

*t*

0

Z

*D*

∆_{y}*G** _{t−s}*(

*x*,

*y*)

*f*(X* _{n}*(s,

*y*))−

*f*(X(s,

*y))*d

*y*ds, and

Λ*n*(*t,x*) :=
Z *t*

0

Z

*D*

*G** _{t−s}*(

*x*,

*y*)

*H*(

*X*

*(*

_{n}*s,y*))

*W** _{n}*(d

*y, ds*)−

*W*(d

*y, ds*)

−
Z *t*

0

Z

*D*

*G*_{t}_{−}* _{s}*(x,

*y)H*˙(X

*n*(s,

*y))*

×

*α**n*(*s,y*)*F*(*X** _{n}*(

*s,y*)) +

*β*

*n*(

*s,y*)

*H*(

*X*

*(*

_{n}*s,y*))

d*yds.* (3.2)

Introduce an auxiliaryF*t**n*-adapted process

*X*_{n}^{−}(t,*x*):=*G*_{t}_{−}_{t}_{n}*x*,*X** _{n}*(t

*n*,·)

, for (t,*x*)∈ O*T*. (3.3)

Recall the localization argument adopted in[7]. For*γ*∈(0, 1)and*p*≥4, define
Φ^{p,}_{n}* ^{γ}*(t):= sup

*s∈[*0,t]k*Y** _{n}*(t,·)k

*p*+ sup

*s*6=s^{0}∈[0,t]

k*Y** _{n}*(s,·)−

*Y*

*(s*

_{n}^{0},·)k

_{p}|*s*−*s*^{0}|* ^{γ}* ,
wherek · k

*p*corresponds to the norm of

*L*

*(D)and for*

^{p}*δ >*0,

*τ*^{δ}* _{n}*:=inf¦

*t>*0; Φ^{p,}_{n}* ^{γ}*(t)≥

*δ*©

∧*T.*

For*M> δ*, define the following events
*A** _{t}*(M−

*δ)*:=

¨

*ω*∈Ω; sup

*s∈[*0,t]k*X*(s,·)k*p*≤*M*−*δ*

«

, (3.4)

*A*^{M}* _{n}*(

*t*) :=

¨

*ω*∈Ω; sup

*s*∈[0,t]k*X** _{n}*(

*s,*·)k

*p*∨ sup

*s*∈[0,t]k*X*(*s,*·)k*p*≤*M*

«

. (3.5)

Then for*t*∈]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]k*X*(s,·)k*p*≤*M*−*δ*

«

∩

¨ sup

*s∈[*0,t]k*X** _{n}*(s,·)−

*X*(s,·)k

*p*≤

*δ*

«

⊆

¨ sup

*s*∈[0,t]k*X*(*s,*·)k*p*≤*M*

«

∩

¨ sup

*s*∈[0,t]k*X** _{n}*(

*s,*·)k

*p*≤

*M*

«

= *A*^{M}* _{n}*(

*t*).

Recall the event ¯Ω^{α}* _{n,t}* defined by (2.6) in Section 2 and that

*α >*2 qlog 2

*Tπ** ^{d}*. For each fixed

*δ >*0 and

*V*∈

*C*

*([0,*

^{γ}*T*];

*L*

*(D)), set*

^{p}k*V*k* _{γ}*,p,

*τ*

^{δ}*:= sup*

_{n}*s∈[*0,T∧τ^{δ}* _{n}*]k

*V*(

*s,*·)k

*p*+ sup

*s6=s*^{0}∈[0,T∧τ^{δ}* _{n}*]

k*V*(s,·)−*V*(s^{0},·)k*p*

|*s*−*s*^{0}|* ^{γ}* .

Then for*q*≥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,T}*i

*c*

= **P**

k*Y** _{n}*k

*,p,*

_{γ}*τ*

^{δ}*≥*

_{n}*δ*,

*A*

*(*

_{T}*M*−

*δ)*∩Ω¯

^{α}*+*

_{n,T}**P**

*A*^{c}* _{T}*(

*M*−

*δ)*

+**P**h

Ω¯^{α}* _{n,T}*i

*c*

≤ *δ*^{−q}**E**

**1**_{A}

*T*(*M*−δ)∩Ω¯^{α}*n,T*k*Y** _{n}*k

^{q}

_{γ}_{,p,}

_{τ}*δ*

*n*

+**P**

*A*^{c}* _{T}*(

*M*−

*δ)*

+**P**h

Ω¯^{α}* _{n,T}*i

*c* , However, Lemma 2.2 and Lemma 3.1 (the latter will be proved below) yield that

**P**

*A*^{c}* _{T}*(M−

*δ)*

+**P**h

Ω¯^{α}* _{n,T}*i

*c*

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

Φ^{p,}_{n}* ^{γ}*(T)→0, in probability, as

*n*→ ∞, (3.7) it suffices to check that there exist

*q*≥

*p*and

*θ >qα*¯(we have set

*γ*=

*α*¯, where ¯

*α*is the exponent presented in Proposition 3.1) such that

(**C1**) ∀*t*∈[0,*T*], lim

*n*→∞**E**h

**1***A*¯^{M}* _{n}*(t)k

*Y*

*(t,·)k*

_{n}

^{q}*i*

_{p}=0;

(**C2**) ∀*s<t*∈[0,*T*], **E**h

**1***A*¯^{M}* _{n}*(t)k

*Y*

*(*

_{n}*t,*·)−

*Y*

*(s,·)k*

_{n}

^{q}*i*

_{p}≤*C*|*t*−*s*|^{1+θ}.
Here the event ¯*A*^{M}* _{n}*(t)is defined by

*A*¯^{M}* _{n}* (t) =

*A*

^{M}*(t)∩Ω¯*

_{n}

^{α}*,*

_{n,t}*t*∈[0,

*T]*, (3.8) and which satisfies the order relation:

*A*¯^{M}* _{n}* (

*t*)⊂

*A*¯

^{M}*(*

_{n}*r*), if

*r*≤

*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 for*p*≥4,*β*∈[p,∞[and*β*∈]p, 6p/(6−*p)*^{+}[(if*d*=3),
**P**

*A** _{T}*(

*M*)

*c*

=**P**

sup

*t*∈[0,T]k*X*(*t,*·)k*p*≥*M*

≤*M*^{−β}**E**

sup

*t*∈[0,T]k*X*(*t*,·)k^{β}_{p}

. Therefore, it remains to prove

**E**

sup

*t*∈[0,T]k*X*(*t,*·)k^{β}_{p}

*<*∞. (3.9)

Define for(*t,x*)∈ O*T*,

*L*_{1}(u)(t,*x*) :=
Z *t*

0

Z

*D*

*G*_{t}_{−}* _{s}*(x,

*y*)[F+

*H](u(s,y*))W(d

*y, ds)*;

*L*_{2}(u)(t,*x*) :=
Z *t*

0

Z

*D*

*G** _{t−s}*(x,

*y*)K(u(s,

*y*))

**h**(s,

*y*)d

*y*ds, and set

*Z*=

*X*−

*L*(

*X*)with

*L*=

*L*

_{1}+

*L*

_{2}. Then

*∂**Z*

*∂**t* + ∆^{2}*Z*−∆*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 any*q,δ*∈]1,∞[and some*γ*^{0},*γ*^{00}∈]0, 1],

(**a**) sup

(t,x)∈O*T*

**E**

|*L(u)(t*,*x*)|^{2qδ}

*<*∞,
(b) **E**

|*L(u)(t*,*x*)−*L(u)(t*^{0},*x*^{0})|^{2q}

≤*C*

|*t*−*t*^{0}|^{γ}^{00}+|*x*−*x*^{0}|^{γ}^{0}*q*

, *q>*1,

then (3.9) holds. So we only need to prove (a) and (b). Note that*K*is bounded and*d*≤3. Then in
light of (A.4),

sup

(*t,x*)∈O*T*

**E**h

*L*_{2}(*u*)(*t*,*x*)
^{2qδ}

i

≤ k*K*k^{2qδ}_{∞} sup

(*t,**x*)∈O*T*

Z *t*

0

Z

*D*

|*G*_{t}_{−}* _{s}*(

*x*,

*y*)

**h**(

*s,y*)|d

*y*ds

2qδ

≤ k*K*k^{2qδ}_{∞} k*h*k^{qδ}_{H}*T*^{(}^{1}^{−}^{d}^{4}^{)}^{q}^{δ}*<*∞.

If set*t>t*^{0}, then by (A.7)–(A.9) in Lemma A.2, we have for*γ*^{0}∈[0, 1−*d/*4[and*γ*^{00}∈[0, 2∧(4−*d*)[,
**E**

|*L*_{2}(u)(t,*x)*−*L*_{2}(u)(t^{0},*x*^{0})|^{2q}

≤ *C*

|*t*−*t*^{0}|^{γ}^{00}+|*x*−*x*^{0}|^{γ}^{0}*q*

.

The estimate of*L*_{1}is similar to that of *L*_{2} (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 withΛ*n*(t,*x*)instead
of*Y** _{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]

**E**h

**1***A*¯^{M}* _{n}*(

*t*)k

*Y*

*(t,·)k*

_{n}

^{q}*i*

_{p}≤*C* sup

*t*∈[0,T]

**E**h

**1***A*¯^{M}* _{n}*(

*t*)kΛ

*n*(t,·)k

^{q}*i*

_{p}, (4.1)

*where*Λ*n*(*t*,*x*)*is defined by*(3.2).

**Proof.** Note that for each*t*∈[0,*T*],
**E**h

k*Y** _{n}*(t,·)k

^{q}*i*

_{p}≤*C***E**

X3

*i=*1

**1***A*¯^{M}* _{n}*(t)kΓ

^{i}*(t,·)k*

_{n}

^{q}*+*

_{p}**1**

*A*¯

^{M}*(t)kΛ*

_{n}*(t,·)k*

_{n}

^{q}

_{p}

,

whereΓ* ^{i}*(

*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}

*+ 1∈[0, 1],*

_{p}**E**
h

**1***A*¯^{M}* _{n}*(t)kΓ

^{1}

*(*

_{n}*t,*·)k

^{q}*i*

_{p}≤ *C*
Z

*D*

**E**h

**1***A*¯^{M}* _{n}*(t)|Γ

^{1}

*(t,*

_{n}*x*)|

*i dx*

^{q}≤ *C***E**

Z *t*

0

Z

*D*

*G*^{2}_{t}_{−}* _{s}*(·,

*y*)1

*A*¯

^{M}*(s)|*

_{n}*Y*

*(s,*

_{n}*y*)|

^{2}d

*yds*

*q*
2

*q*
2

≤ *C***E**

Z *t*

0

(*t*−*s*)^{4r}^{d}^{−}^{d}^{2}**1***A*¯^{M}* _{n}*(

*s*)k

*Y*

*(*

_{n}*s,*·)k

^{2}

*ds*

_{p}

*q*
2

≤ *C*

Z *t*

0

(*t*−*s*)^{4r}^{d}^{−}^{d}^{2}ds

*q*−2
2 Z *t*

0

(*t*−*s*)^{4r}^{d}^{−}^{d}^{2} sup

*r*∈[0,s]

**E**h

**1***A*¯^{M}* _{n}*(

*r*)k

*Y*

*(*

_{n}*r,*·)k

^{q}*i ds*

_{p} . Similarly, we have

**E**h

**1***A*¯^{M}* _{n}*(t)kΓ

^{2}

*(*

_{n}*t,*·)k

^{q}*i*

_{p}≤ *C*k*h*k^{q}_{H}**E**

Z *t*

0

Z

*D*

*G*^{2}_{t}_{−}* _{s}*(·,

*y*)

**1**

*A*¯

^{M}*(s)|*

_{n}*Y*

*(s,*

_{n}*y*)|

^{2}d

*y*ds

*q*
2

*q*
2

≤ *C*

Z *t*

0

(t−*s)*^{4r}^{d}^{−}^{d}^{2}ds

^{q−}

2
2 Z *t*

0

(t−*s)*^{4r}^{d}^{−}^{d}^{2} sup

*r∈[*0,s]

**E**h

**1***A*¯^{M}* _{n}*(r)k

*Y*

*(*

_{n}*r,*·)k

^{q}*i ds*

_{p}
.
As forΓ^{3}* _{n}*(t,

*x*), using (A.13) with

^{1}

*r*2 =^{1}* _{q}*−

^{1}

*+1∈[0, 1],*

_{ρ}**E**h

**1***A*¯^{M}* _{n}*(t)kΓ

^{3}

*(t,·)k*

_{n}

^{q}*i*

_{p}≤ *C***E**

**1***A*¯^{M}* _{n}*(t)

Z *t*

0

Z

*D*

∆_{y}*G(t*−*s,*·,*y*)[*f*(X* _{n}*(s,·))−

*f*(X(s,·))]d

*y*ds

*q*

*q*

≤ *C***E**

1*A*¯^{M}* _{n}*(t)

Z *t*

0

(t−*s)*^{4r}^{d}^{2}^{−}^{d+2}^{4} k*f*(X*n*(s,·))−*f*(X(s,·))k* _{ρ}*ds

*q*

. (4.2)

Note that*u*(*s,*·),*v*(*s,*·)∈*L** ^{p}*(

*D*), for each

*s*∈[0,

*T*], we have for

*ρ*=

_{3}

*, k*

^{p}*u*(

*s,*·)−

*v*(

*s,*·)k

*≤*

_{ρ}*C*k

*u*(

*s,*·)−

*v*(

*s,*·)k

*p*,

k*u*^{2}(s,·)−*v*^{2}(s,·)k* _{ρ}* ≤

*C*k

*u(s,*·)−

*v(s,*·)k

**

_{p}k*u(s,*·)k* _{p}*+k

*v(s,*·)k

**

_{p}k*u*^{3}(s,·)−*v*^{3}(s,·)k* _{ρ}* ≤

*C*k

*u(s,*·)−

*v(s,*·)k

*p*

×h

k*u*(*s,*·)k^{2}* _{p}*+k

*v*(

*s,*·)k

^{2}

*+k*

_{p}*u*(

*s,*·)k

*p*k

*v*(

*s,*·)k

*p*

i . Hence from (4.2), it follows that

**E**h

**1***A*¯^{M}* _{n}*(

*t*)kΓ

^{3}

*(t,·)k*

_{n}

^{q}*i*

_{p}≤ *C***E**

Z *t*

0

(t−*s)*^{4r}^{d}^{2}^{−}^{d+2}^{4} **1***A*¯^{M}* _{n}*(s)k

*Y*

*(s,·)k*

_{n}*ds*

_{p}*q*

≤ *C*

Z *t*

0

(*t*−*s)*^{4r}^{d}^{2}^{−}^{d+2}^{4} ds

*q−*1Z *t*

0

(t−*s)*^{4r}^{d}^{2}^{−}^{d+2}^{4} sup

*r∈[*0,s]

**E**h

**1***A*¯^{M}* _{n}*(r)k

*Y*

*(r,·)k*

_{n}

^{q}*i ds*

_{p} . Note that the following equivalent relations holds:

*d*
4r − *d*

2+1*>*0 ⇔ 1
*q* *>* 1

*p* −1
6,
*d*

4r_{2}− *d*+2

4 +1*>*0 ⇔ 1
*q* *>* 1

*p* +1
2− 2

*d*.

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

Recall theF*t** _{n}*-adapted process

*X*

^{−}

*(*

_{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}*>*0

*such that*

sup

(t,x)∈O*T*

**E**h

**1***A*¯^{M}* _{n}*(

*t*)|

*X*

*(t,*

_{n}*x*)−

*X*

_{n}^{−}(

*t,x*)|

*i*

^{q}≤*C*2^{−nqι}, (4.3)

*whereι*:= ^{1}_{2}(1− ^{d}_{4})*.*

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

*X** _{n}*(t,

*x*)−

*X*

^{−}

*(t,*

_{n}*x*) =

4

X

*k=*1

*T*_{n}* ^{k}*(t,

*x*), where

*T*_{n}^{1}(*t,x*) :=
Z *t*

*t*_{n}

Z

*D*

*G*_{t}_{−}* _{s}*(x,

*y*)F(X

*n*(s,

*y))W*(d

*y, ds)*,

*T*_{n}^{2}(*t,x*) :=
Z *t*

*t*_{n}

Z

*D*

*G*_{t}_{−}* _{s}*(

*x*,

*y*)

*H*(

*X*

*(*

_{n}*s,y*))

*W*

*(d*

_{n}*y, ds*),

*T*_{n}^{3}(*t,x*) :=
Z *t*

*t*_{n}

Z

*D*

∆*y**G** _{t−s}*(

*x*,

*y*)

*f*(

*X*

*(*

_{n}*s,y*))dyds,

*T*_{n}^{4}(*t,x*) :=
Z *t*

*t*_{n}

Z

*D*

*G** _{t−s}*(

*x*,

*y*)

*K*

*(*

_{n}*s,y*)dyds,

and

*K** _{n}*(

*s,y*) =

*K*(

*X*

*(*

_{n}*s,y*))

**h**(

*s,y*)

−(F*H*˙)(X* _{n}*(s,

*y*))α

*(s,*

_{n}*y*)−(H

*H)(X*˙

*(s,*

_{n}*y))β*

*(s,*

_{n}*y*). From (2.4) and (2.5), it follows that for

*h*∈ Hb,

sup

(s,*y)∈O**T*

|*K** _{n}*(

*s,y*)| ≤

*C n*

*. (4.4)*

^{d}On the other hand, using (A.4) and the boundedness of*F,*
**E**

|*T*_{n}^{1}(t,*x*)|* ^{q}*

= **E**

Z *t*

*t**n*

Z

*D*

*G** _{t−s}*(x,

*y*)F(X

*(s,*

_{n}*y*))W(d

*y, ds)*

*q*

≤ *C***E**

Z *t*

*t*_{n}

Z

*D*

*G*^{2}_{t}_{−}* _{s}*(

*x*,

*y*)F

^{2}(X

*(s,*

_{n}*y*))d

*y*ds

*q/*2

≤ *C*

Z *t*

*t*_{n}

Z

*D*

*G*^{2}_{t}_{−}* _{s}*(x,

*y*)dyds

*q**/*2

≤ *C*

Z *t*

*t*_{n}

(t−*s)*^{−}^{d}^{4}ds

*q/2*

≤ *C*2^{−}^{1}^{2}^{nq(}^{1}^{−}^{d}^{4}^{)}. (4.5)

Further, the Hölder inequality, Lemma 2.1 and the boundedness of*H* jointly imply that
**E**

|*T*_{n}^{2}(t,*x*)|* ^{q}*

= **E**

Z *t*

*t*_{n}

Z

*D*

*G*_{t}_{−}* _{s}*(

*x*,

*y*)H(X

*n*(s,

*y*))W

*n*(dy, ds)

*q*

≤ *C***E**

Z *t*

*t*_{n}

Z

*D*

|*G** _{t−s}*(

*x,y*)||

*W*˙

*(*

_{n}*y,s)|*d

*y*ds

*q*

≤ *C***E**

Z *t*

*t*_{n}

Z

*D*

|*G*_{t}_{−}* _{s}*(x,

*y)||W*˙

*(*

_{n}*y,s)|*

*dyds*

^{q}

·

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*)|d

*yds*

*q*

sup

(*t,x*)∈O*T*

**E**

|*W*˙* _{n}*(

*x,t*)|

**

^{q}≤ *C n*^{dq}^{2}2^{nq}^{2}|*t*−*t** _{n}*|

^{q}≤ *C n*

*dq*

22^{−}^{2}* ^{nq}*. (4.6)

Note that*f* is a polynomial of degree 3. Then by virtue of (A.13) with*κ*= _{∞}^{1} −^{3}* _{p}*+1=1−

^{3}

*∈[0, 1],*

_{p}**E**h

**1***A*¯^{M}* _{n}*(

*t*)|

*T*

_{n}^{3}(

*t,x*)|

*i*

^{q}= **E**

1*A*¯^{M}* _{n}*(t)

Z *t*

*t**n*

Z

*D*

∆_{y}*G** _{t−s}*(

*x*,

*y*)

*f*(X

*(s,*

_{n}*y*))d

*y*ds

*q*

≤ **E**

Z *t*

*t*_{n}

(t−*s)*^{d}^{4}^{κ−}^{d+2}^{4} **1***A*¯^{M}* _{n}*(

*s*)k

*f*(X

*n*(s,·))k

^{p}3ds

*q*

≤ **E**

Z *t*

*t**n*

(t−*s)*^{d}^{4}^{(1}^{−}^{3}^{p}^{)−}^{d+2}^{4} **1***A*¯^{M}* _{n}*(s)

h

k*X** _{n}*(s,·)k

*+k*

_{p}*X*

*(s,·)k*

_{n}^{2}

*+k*

_{p}*X*

*(s,·)k*

_{n}^{3}

*i ds*

_{p}

*q*

≤ *C*

Z *t*

*t*_{n}

(t−*s)*^{−(}^{1}^{2}^{+}^{3d}^{4p}^{)}ds

*q*

≤ *C*2^{−nq(}

1

2−^{3d}_{4p}). (4.7)

Thanks to (4.4), we conclude that
**E**

|*T*_{n}^{4}(*t,x*)|* ^{q}*

= **E**

Z *t*

*t*_{n}

Z

*D*

*G*_{t}_{−}* _{s}*(x,

*y)K*

*n*(s,

*y*)d

*y*ds

*q*

≤ *C n*^{dq}

Z *t*

*t*_{n}

Z

*D*

|*G** _{t−s}*(

*x*,

*y*)|d

*y*ds

*q*

≤ *C n** ^{dq}*|

*t*−

*t*

*|*

_{n}

^{q}≤ *C n** ^{dq}*2

^{−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}*>*0*such that for q*≥*p*≥4,
sup

*t*∈[0,T]

**E**h

**1***A*¯^{M}* _{n}*(

*t*)k

*X*

*(*

_{n}*t,*·)−

*X*

_{n}^{−}(

*t,*·)k

^{q}*i*

_{p}≤*C** _{M}*2

^{−nqι}, (4.9)

*whereι*= ^{1}_{2}(1−^{d}_{4}). In fact, for the proof of(4.9), the only estimate that needs to be checked is that of
*T*_{n}^{4}*. Apply*(A.13)*withκ*= ^{1}* _{p}*−

^{3}

*+1=1−*

_{p}^{2}

*∈[0, 1]*

_{p}*to T*

_{n}^{4}

*and we can get*(4.9).

Next we prove a useful lemma, which will be used frequently later. For(*t*,*x*),(*s,y*)∈ O*T*, 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) Let

*V*be anF= (F

*t*)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}

2*n* ∧t

*kT*
2*n*∧t

Z

*D*

*η(t,s,x*,*y*)

(r,*z)V*(r,*z)W** _{n}*(dz, dr), (4.12)

with*k*=0, 1, . . . , 2* ^{n}*−1. Then we have