getdoc8953. 341KB Jun 04 2011 12:04:38 AM

El e c t ro n ic

Jo ur

n a l o

f

P r

o

b a b i_{l 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

1

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

2

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

3

School 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).

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)

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

H =

¨

h(t,x) =

Z t

0

Z

Qd i=1[0,xi]

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

h_∈L2([0,T]_×D)

«

,

and the corresponding norm by

kh_kH =

sZ T

0

Z

D

|h(s,y)_|2_{dyds, for all h∈ H.}

Let_Hb 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∈Lp(D)forp_≥4, andψis̺∈]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],Lp(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 Cα¯, ¯β([0,T]_×D)of the lawP_◦u−1is the closure ofS_H.

(b) Let p ≥ 4. Then for α¯ ∈]0, min_{1 2(1−

d

4), ̺

4}[, the topological support supp(P◦u

−1_{) in}

Cα¯([0,T],Lp(D))of the lawP◦u−1is the closure ofS_H.

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 tn=

”

t_n−T2−n—∨0.

Let k := (k1, . . . ,kd) ∈ Idn := {0, 1, . . . ,n−1}d. Define a partition (△j,k)j=0,1,...,2n−1_,k∈Id

n of OT :=

[0,T]_×Dby

△j,k=Dk×]j T2−n,(j+1)T2−n],

where D_k = Qd_j=1]k_jπn−1,(k_j+1)πn−1]. For x_{j ∈}]k_jπn−1,(k_j+1)πn−1]with j = 1, . . . ,d, we

set Dk(x) = Qd

j=1]kjπn−1,(kj +1)πn−1]. Further, for each (t,x) ∈ OT, we define the following

difference approximation to ˙W by

˙

W_n(x,t) =

( _W(△

j−1,k)

|△j−1,k| , (x,t)∈ △j,k, j=1, . . . , 2

n

−1, k_∈Id_n, 0, (x,t)_{∈ △0,k}, k∈Id_n,

(2.1)

where _△j,k= Tπd(nd2n)−1 is the volume of the partition_△j,k for each j = 0, 1, . . . , 2n₋1 and k∈Id_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},

Xn(t,x) = Gt∗ψ(x)

+

Z t

0

Z

D

G_t−s(x,y)F(X_n(s,y))W(dy, ds) +H(X_n(s,y))W_n(dy, ds)

+

Z t

0

Z

D

Gt−s(x,y)

h

K(Xn(s,y))h(s,y)

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

i

dyds

+

Z t

0

Z

D

∆_yGt−s(x,y)f(Xn(s,y))dyds, (2.2)

and

X(t,x) = Gt∗ψ(x) +

Z t

0

Z

D

Gt−s(x,y)[F+H](X(s,y))W(dy, ds)

+

Z t

0

Z

D

+

Z t

0

Z

D

∆_yGt−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,

α_n(t,x) := nd2n(Tπd)−1

Z t_n

tn

Z

Dk(x)

G_t−s(x,y)dyds,

βn(t,x) := nd2n(Tπd)−1

Z t

t_n

Z

Dk(x)

Gt−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 nd, and (2.4)

sup (t,x)∈OT

|βn(t,x)| ≤C nd. (2.5)

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

sup

x∈D|

αn(t,x)| ≤ C nd2nmax

k∈Id n

(

1_Dk(x)

Z t_n

tn

Z

Dk

|G_t−s(x,y)_|dyds

)

≤ C nd2n_|t_n−t_n|

≤ C nd,

and

sup

x∈D|

βn(t,x)| ≤C nd2n|t−tn| ≤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—_≤Cpn d p

22

np

Proof. By virtue of the definition (2.1),

sup (t,x)∈OT

E”|_W˙_n(x,t)_|p—

= sup

(t,x)∈OT

E

2_Xn−1

j=1

X

k∈Id n

W(_△j−1,k)

|△j−1,k| 1_△j

−1,k(x,t)

p

≤ Cpmax

  E

W(_△j−1,k)

|△j−1,k|

p

; j=1, . . . , 2n−1, k∈Id_n

  .

Note that for each j=1, . . . , 2n₋1 andk_∈Id_n,

W(_△j−1,k)

|△j−1,k| ∼

N(0,_|△j−1,k|−1).

For any random variableZ∼N(0,σ2), it holds that

E_|Z|p= _p1 π

p

2pσ2p_Γ

_p

2+ 1 2

,

whereΓdenotes the Gamma function. This yields that

sup (t,x)∈OT

E”_|W˙_n(x,t)_|p—_≤C_pmaxnp_|△j−1,k|−p; j=1, . . . , 2n₋1, k_∈Id_no,

and which proves the lemma. ƒ

Letn_∈Nbe fixed. Forα >0 and t_∈]0,T], we now define an event ¯Ωα_n,t by

¯ Ωα_n,t=

¨

ω∈Ω; sup (s,y)∈[0,t]×D

_W˙_{(y,s;ω)≤α}nd2n2

«

. (2.6)

For this event, we have

Lemma 2.2. If chooseα >2

q

log 2

Tπd, then

lim

n→∞P

h

¯ Ωα_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),

PhΩ¯α_n,Tic

‹

= P

‚

max (j,k)∈{1,...,2n_−1}×Id

n

¨

W(_△j−1,k) |△j−1,k|

«

≥αnd22n

Œ

≤ nd2nP



|Z_{| ≥}α

p

= nd2nP

‚

|Z|2

4 ≥

α2Tπd

4 n

d

Œ

≤ nd2nexp

–

−α 2_Tπd

4 n

d

™

E

–

exp

‚

|Z|2 4

Ϊ

. (2.7)

Note thatE

h

exp

|Z|2 4

i

=p2. Then (2.7) further yields that

0≤P

h

¯ Ωα_n,Tic

‹

≤ p2ndexp

–

nlog 2− α 2_Tπd

4 n

d

™

≤ p2ndexp

–

nd

‚

log 2−α 2_Tπd

4

Ϊ

→ 0, as n_{→ ∞}, (2.8)

ifα >2

q

log 2

Tπd. Thus the proof of the lemma is complete. ƒ

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],Lp(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 Xn

converges in probability to X in Cα¯, ¯β([0,T]_×D).

(ii) Let p_≥4. Then forα¯∈]0, min_{1 2(1−

d

4), ̺

4}[, the sequence Xn converges in probability to X in

Cα¯([0,T],Lp(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

Yn(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

Γ3_n(t,x) :=

Z t

0

Z

D

∆_yGt−s(x,y)

f(Xn(s,y))− f(X(s,y))

dyds,

and

Λ_n(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˙(X_n(s,y))

×αn(s,y)F(Xn(s,y)) +βn(s,y)H(Xn(s,y))

dyds. (3.2)

Introduce an auxiliaryFtn-adapted process

X_n−(t,x):=G_t−t

n x,Xn(tn,·)

, for (t,x)_{∈ OT}. (3.3)

Recall the localization argument adopted in[7]. Forγ∈(0, 1)andp≥4, define

Φp_n,γ(t):= sup

s∈[0,t]k

Yn(t,·)kp+ sup s6=s′_∈[0,t]

kY_n(s,·)₋Y_n(s′,·)_kp |s₋s′_|γ ,

where_{k · kp} corresponds to the norm ofLp(D)and forδ >0,

τ_nδ:=inf¦t>0; Φ_np,γ(t)_≥δ©∧T.

ForM> δ, define the following events

A_t(M₋δ) :=

¨

ω∈Ω; sup

s∈[0,t]k

X(s,_·)_kp≤M₋δ

«

, (3.4)

AM_n(t) :=

¨

ω∈Ω; sup

s∈[0,t]k

X_n(s,_·)_kp∨ sup

s∈[0,t]k

X(s,_·)_kp≤M

«

. (3.5)

Then fort_∈]0,T],

A_t(M₋δ)∩ {t_≤τδ_n} ⊆AM_n(t). (3.6)

In fact, from the inequality_|y| ≤ |x−y|+_|x|, it follows that

A_t(M₋δ)∩ {t_≤τδ_n}

⊆

¨

sup

s∈[0,t]k

X(s,_·)_kp≤M₋δ

«

∩

¨

sup

s∈[0,t]k

X_n(s,_·)₋X(s,_·)_kp≤δ

«

⊆

¨

sup

s∈[0,t]k

X(s,_·)_kp≤M

«

∩

¨

sup

s∈[0,t]k

X_n(s,_·)_kp≤M

«

= AM_n(t).

Recall the event ¯Ωα_n,t defined by (2.6) in Section 2 and thatα >2

q

log 2

Tπd. For each fixedδ >0 and V ∈Cγ([0,T];Lp(D)), set

kV_kγ,p,τδ

n := sup

s∈[0,T∧τδ n]

kV(s,_·)_kp+ sup

s6=s′_∈[0,T∧τδ n]

Then forq_≥1,

P€Φp_n,γ(T)_≥δŠ

≤ PΦp_n,γ(T)_≥δ, A_T(M−δ)∩Ω¯α_n,T+P€Ac_T(M−δ)Š+P

h

¯ Ωα_n,Tic

‹

= P_kY_nkγ,p,τδ

n≥δ, AT(M−δ)∩

¯

Ωα_n,T+P€Ac_T(M₋δ)Š+P

h

¯ Ωα_n,Tic

‹

≤ δ−qE

•

1_A

T(M−δ)∩Ω¯αn,TkYnk q

γ,p,τδ n

˜

+P€Ac_T(M₋δ)Š+PhΩ¯α_n,Tic

‹

,

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

P€Ac_T(M₋δ)Š+PhΩ¯α_n,Tic

‹

→0, asM _{→ ∞}, n_{→ ∞}.

Therefore, by Lemma A.1 in[7]and (3.6), in order to prove

Φp_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→∞E

h

1¯_AM

n(t)kYn(t,·)k q p

i

=0;

(C2) _∀s<t_∈[0,T], Eh1¯_AM

n(t)kYn(t,·)−Yn(s,·)k q p

i

≤C_|t₋s_|1+θ.

Here the event ¯AM_n(t)is defined by

¯

AM_n (t) =AM_n(t)_∩Ω¯α_n,t, t_∈[0,T], (3.8)

and which satisfies the order relation:

¯

AM_n (t)_⊂A¯M_n(r), ifr_≤t.

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

lim

M→∞P

€

A_T(M)cŠ=0.

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

P€A_T(M)cŠ=P

‚

sup

t∈[0,T]k

X(t,_·)_kp≥M

Œ

≤M−βE

–

sup

t∈[0,T]k

X(t,_·)_kβ_p

™

.

Therefore, it remains to prove

E

‚

sup

t∈[0,T]k

X(t,_·)_kβ_p

Œ

<∞. (3.9)

Define for(t,x)_{∈ OT},

L₁(u)(t,x) :=

Z t

0

Z

D

L2(u)(t,x) :=

Z t

0

Z

D

Gt−s(x,y)K(u(s,y))h(s,y)dyds,

and setZ=X₋L(X)withL=L₁+L₂. 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 anyq,δ∈]1,∞[and someγ′,γ′′∈]0, 1],

(a) sup

(t,x)∈OT

E”|L(u)(t,x)_|2qδ—<∞,

(b) E”|L(u)(t,x)₋L(u)(t′,x′)_|2q—_≤C”|t−t′|γ′′+_|x−x′|γ′—q, 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

EhL₂(u)(t,x)2qδi _{≤ k}K_k2_∞qδ sup (t,x)∈OT

Z t

0

Z

D

|G_t−s(x,y)h(s,y)_|dyds

2qδ

≤ kK_k2_∞qδ_kh_kHqδT(1−d4)qδ<∞.

If sett>t′, then by (A.7)–(A.9) in Lemma A.2, we have forγ′∈[0, 1₋d/4[andγ′′∈[0, 2_∧(4₋d)[,

E”|L2(u)(t,x)−L2(u)(t′,x′)|2q

—

≤ C”|t−t′|γ′′+_|x−x′|γ′—q.

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

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 ofYn(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] Eh1¯_AM

n(t)kYn(t,·)k q p

i

≤C sup

t∈[0,T] Eh1A¯M

n(t)kΛn(t,·)k q p

i

, (4.1)

Proof. Note that for eacht_∈[0,T],

Similarly, we have

ku3(s,_·)₋v3(s,_·)_{kρ ≤} C_ku(s,_·)₋v(s,_·)_kp

×hku(s,_·)_k2_p+_kv(s,_·)_kp2+_ku(s,_·)_kpkv(s,_·)_kpi.

Hence from (4.2), it follows that

Eh1¯_AM n(t)kΓ

3

n(t,·)k q p

i

≤ CE

–Z t

0 (t−s)

d

4r₂− d+2

4 ₁ ¯

AM

n(s)kYn(s,·)kpds

™q

≤ C

–Z t

0

(t₋s)

d

4r₂− d+2

4 _ds

™q−1–Z t

0 (t₋s)

d

4r₂− d+2

4 _sup

r∈[0,s] Eh1A¯M

n(r)kYn(r,·)k q p

i

ds

™

.

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. ƒ

Recall the_Ft

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

Eh1¯_AM

n(t)|Xn(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_nk(t,x),

where

T_n1(t,x) :=

Z t

tn

Z

D

G_t−s(x,y)F(X_n(s,y))W(dy, ds),

T_n2(t,x) :=

Z t

tn

Z

D

G_t−s(x,y)H(X_n(s,y))W_n(dy, ds),

T_n3(t,x) :=

Z t

tn

Z

D

∆_yG_t−s(x,y)f(X_n(s,y))dyds,

T_n4(t,x) :=

Z t

tn

Z

D

and

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

E”|T_n1(t,x)_|q— = E

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

= E

Thanks to (4.4), we conclude that

E”_|T_n4(t,x)_|q— = E

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

Lemma 4.3. Let V :Ω_{× OT →}Rbe anF-predictable process. If for each p_∈[1,_∞[, there exist some

Proof. As in (4.10), define

λk_n(V)(t,x,y):= by (4.13) and Lemma 2.1,

≤ C nd p+2ap2np−2npαˆ−n

Next we turn to the time increment. Set

λk_n(V)(t,s,x):=

with the definition

Ak_2,n(V)(t,s,x) =0, whenever [(k+1)T2−n∧t]<[kT2−n∨s].

Then from (A.4) and the Hölder inequality, it follows that

≤ C nd+2a2−n(2αˆ−1+ 1

p)

  

Z T

2n(k+1)∧t

T k

2n∨s

(t−r)−d4(γβ−1)dr

  

2

γβ

≤ C nd+2a2−n(2αˆ−1+ 1

p)₂−n(

2

γβ−1)

–Z t

s

(t₋r)−d4(γβ−1)dr

™

≤ C nd+2a2−n(2αˆ−2+ 1

p+

2

γβ)_|t−s|−d4(γβ−1)+1. (4.22)

Thus from (4.17), (4.20) and (4.22), we get (4.14). Hence the proof of the lemma is complete. ƒ

Remark 4.2. From the proof of Lemma 4.3, the conclusion of the lemma still holds if we replace

λk

n(V)(t,s,x,y)byE

•

λk

n(V)(t,s,x,y)|F(k−1)T

2n

˜

.

Lemma 4.4. For(s,y)_{∈ OT}, it holds that

E



_W˙_n(y,s)

Z s

sn

Z

D

Gs−r(y,z)F(X−n(r,z))W(dz, dr)|Fsn

 

= (Tπd)−1nd2n

Z s_n

sn

Z

Dk(y)

Gs−r(y,z)F(Xn−(r,z))dzdr.

Proof. Foru_≥s_n, set

N_u(s,y) := (Tπd)−1nd2nW(D_k(y)_×]s_n,u]),

M_u(s,y) :=

Z u

sn

Z

D

G_s−r(y,z)F(X_n−(r,z))W(dz, dr).

Then the martingale property of_{Mu(s,y);u≥sn}and Itô formula jointly imply that

E



_W˙_n(y,s)

Z s

sn

Z

D

Gs−r(y,z)F(X−n(r,z))W(dz, dr)|Fsn

 

= (Tπd)−1nd2nE

h

Ns_n(s,y)Ms(s,y)|Fsn

i

= (Tπd)−1nd2nEhEhN_s

n(s,y)Ms(s,y)|Fsn

i

Fsn

i

= (Tπd)−1nd2nEhN_s

n(s,y)E[Ms(s,y)|Fsn]Fsn

i

= (Tπd)−1nd2nE

h

N_s

n(s,y)Msn(s,y)|Fsn

i

= (Tπd)−1nd2n

Z s_n

sn

Z

Dk(y)

G_s−r(y,z)F(X_n−(r,z))dzdr,

follows from the fact thatF(X−_n(r,z))isFsn-measurable whenr≤s. This proves the lemma. ƒ

Lemma 4.5. Let q _≥ p _≥6. Then for α¯ ∈]0, min_{1₂(1₋ d₄),̺_4}[ andβ¯ ∈]0, min_{2₋ d₂,̺}[, there exists a C>0such that

Eh1_A¯M

n(t)∩A¯Mn(s)

Xn(t,x)−Xn(s,y)

qi

≤C ndq

h

|t−s|α¯+_|x−y|β¯iq,

with(t,x),(s,y)_{∈ OT}. In addition, if q _≥ p_≥ 4, then forα¯ ∈]0, min_{1 2(1−

d

4), ̺

4}[, there exists a

C >0such that

E

h

1_A¯M

n(t)∩A¯Mn(s)

Xn(t,·)−Xn(s,·)

q

p

i

≤C ndq|t−s|α¯q.

Proof. Recall (2.2), (4.11) and (4.12). We have for(t,x),(s,y)_{∈ OT},

X_n(t,x)₋X_n(s,y) = 5

X

i=1

J_i(s,t,x,y),

where

J₁(t,s,x,y) := G_t∗ψ(x)₋G_s∗ψ(y),

J₂(t,s,x,y) :=

Z t

0

Z

D

[η(t,s,x,y)](r,z)hF(X_n(r,z))W(dz, dr)

+H(X_n−(r,z))W_n(dz, dr)i,

J3(t,s,x,y) :=

Z t

0

Z

D

[η(t,s,x,y)](r,z)h(H(Xn(r,z))−H(Xn−(r,z)))Wn(dz, dr)

−H˙(X_n(r,z))αn(r,z)F(Xn(r,z)) +βn(r,z)H(Xn(r,z))

dzdri,

J4(t,s,x,y) :=

Z t

0

Z

D

[η(t,s,x,y)](r,z)K(Xn(r,z))˙h(r,z)dzdr,

J₅(t,s,x,y) :=

Z t

0

Z

D

˜

η(t,s,x,y)(r,z)f(X_n(r,z))dzdr.

Note that the initial functionψ(x)is̺-Hölder continuous inx _∈Rd. Then by Lemma 2.2 in[6],

E”J1(s,t,x,y)

q—

≤C”|t−s|̺/4+_|x−y|̺—q. (4.23)

SinceF,H are bounded, by Burkhölder’s inequality and Lemma A.2, there existγ′∈]0, 4₋d[,γ′≤2 andγ′′∈]0, 1−d/4[such that

E_|J₂(t,s,x,y)_|q_≤C”_|x₋y_|γ′+_|t₋s_|γ′′— q

2_{. (4.24)}

On the other hand, sinceK is bounded, the Schwarz’s inequality yields that forh_{∈ Hb},

E_|J₄(t,s,x,y)_|q_≤C”_|x₋y_|γ′+_|t₋s_|γ′′— q

2_{, (4.25)}

where γ′,γ′′ are presented in (4.24). Similar as in the proof of (4.7), by (A.14) and (A.15) with 1

r =

1

∞−

3

p+1∈[0, 1], we have forα∈]0, 1/2−

3d

4p[andβ∈]0, min{2−

3d p ,

2

d, 1}],

Eh1¯_AM

n(t)∩A¯Mn(s)|J5(t,s,x,y)| qi

and using (A.14) withκ= 1

p−

3

p+1=1−

2

p∈[0, 1], forα∈]0, 1/2− d

2p[,

Eh1A¯M

n(t)∩A¯Mn(s)kJ5(t,s,·,·)k q p

i

≤C[_|t₋s_|α]q. (4.27)

In the following, we are going to estimateJ3(t,s,x,y). Using the Taylor expansion of H at point

X−_n(t,x),

H(Xn(t,x))−H(Xn−(t,x)) =H˙(Xn−(t,x))

”

Xn(t,x)−X−n(t,x)

—

+ρn(t,x), (4.28)

and

ρn(t,x)≤C|Xn(t,x)−Xn−(t,x)|

2_.

Recall the aboveJ₃, and we have

J3(t,s,x,y) = 5

X

i=1

J₃i(t,s,x,y),

where

J₃1(t,s,x,y) :=

Z t

0

Z

D

[η(t,s,x,y)](r,z)hT_n1(r,z)H˙(X_n−(r,z))W˙_n(z,r)

−_H˙(X_n(r,z))α_n(r,z)F(X_n(r,z))idzdr,

J₃2(t,s,x,y) :=

Z t

0

Z

D

[η(t,s,x,y)](r,z)hT_n2(r,z)H˙(X_n−(r,z))W˙n(z,r)

−H˙(X_n(r,z))βn(r,z)F(Xn(r,z))

i

dzdr,

J₃3(t,s,x,y) :=

Z t

0

Z

D

[η(t,s,x,y)](r,z)H˙(X_n−(r,z))T_n3(r,z)W˙n(z,r)dzdr,

J₃4(t,s,x,y) :=

Z t

0

Z

D

[η(t,s,x,y)](r,z)H˙(X_n−(r,z))T_n4(r,z)W˙_n(z,r)dzdr,

J₃5(t,s,x,y) :=

Z t

0

Z

D

[η(t,s,x,y)](r,z)ρn(r,z)W˙n(z,r)dzdr.

Next we decomposeJ₃1(t,s,x,y)as follows:

J₃1(t,s,x,y) = 5

X

j=1

J₃1,j(t,s,x,y),

with

J₃1,1(t,s,x,y) =

Z t

0

Z

D

[η(t,s,x,y)](r,z)_H˙(X_n−(r,z))_W˙_n(z,r)

×

 

Z r

rn

Z

D

Gr−u(z,v)

”

F(Xn(u,v))−F(Xn−(u,v))

—

W(dv, du)

J₃1,2(t,s,x,y) =

Z t

0

Z

D

[η(t,s,x,y)](r,z)_H˙(X_n−(r,z))

×

– Z r

rn

Z

D

G_r−u(z,v)F(X_n−(r,z))W(dv, du)W˙_n(z,r)

−E

 

Z r

rn

Z

D

Gr−u(z,v)F(Xn−(u,v))W(dv, du)W˙n(z,r)|Frn

 

™

dzdr

J₃1,3(t,s,x,y) =

Z t

0

Z

D

[η(t,s,x,y)](r,z)_H˙(X_n−(r,z))

×

–

E

 

Z r

rn

Z

D

G_r−u(z,v)F(X−_n(u,v))W(dv, du)_W˙_n(z,r)_|Fr

n

 

−2nn−dT−1π−d

Z r_n

rn

Z

Dk(z)

G_r−u(z,v)F(X_n−(u,v))dvdu

™

dzdr,

J₃1,4(t,s,x,y) = 2nT−1n−dπ−d

Z t

0

Z

D

[η(t,s,x,y)](r,z)H˙(X_n−(r,z))

×

 

Z r_n

rn

Z

Dk(z)

G_r−u(z,v)F(X_n−(u,v))dvdu₋αn(r,z)F(Xn(r,z))

 dzdr,

J₃1,5(t,s,x,y) =

Z t

0

Z

D

[η(t,s,x,y)](r,z)”H˙(X_n−(r,z))₋H˙(X_n(r,x))—

×α_n(r,z)F(X_n(r,z))dzdr.

We first estimate the termJ₃1,1. Define

V(r,z) = 1¯_AM n(r)

˙

H(X_n−(r,z))

×

Z r

rn

Z

D

G_r−u(z,v)”F(X_n(u,v))₋F(X_n−(u,v))—W(dv, du). (4.29)

Then

J₃1,1(t,s,x,y) =

Z t

0

Z

D

[η(t,s,x,y)](r,z)V(r,z)W˙_n(dz, dr)

=

[2n_t/T]−1

X

k=0

λk_n(V)(t,s,x,y), (4.30)

where λk_n(V)(t,s,x,y) is defined by (4.12). By virtue of the Burkhölder inequality, (A.16) with κ= 2

q−

2

q+1=1 and Lemma 4.2,

Eh_kV(r,_·)_kqqi _≤ C2−nq(1−d4) sup

r∈[0,T] Eh1A¯M

n(r)kXn(r,·)−X − n(r,·)k

q q

i

Then using Lemma 4.3, there existθ′<4₋dandθ′≤2,θ <1₋d₄ such that

Applying the discrete Burkhölder inequality and Jensen’s inequality to conclude that

E”_|J₃1,2(t,s,x,y)_|q—

Also using Lemma 4.3,

E

γβ >0. On the other hand, Lemma 4.4 yields that

= 2nT−1ndπ−d

Z rn

rn

Z

Dk(z)

Gr−u(z,v)F(X−n(u,v))dvdu.

This implies that

J₃1,3(t,s,x,y)_≡0.

Since ˙H andF are bounded, from (2.4), (A.4) and (A.7)–(A.9), it follows that

J₃1,4(t,s,x,y)

q

≤ C ndq

–Z t

0

Z

D

_[η(t,s,x,y)](r,z)dzdr

™q

×



2n sup (r,z)∈OT

Z r_n

rn

Z

Dk(z)

|G_r−u(z,v)_|dvdu

 

q

≤ C ndq”|t−s|γ′′+_|x−y|γ′—q, (4.37)

where 0< γ′ < 2₋ d₂ and 0 < γ′′ < 1₂(1₋ d₄). From (2.4), Lemma 4.2 and (A.11)–(A.12) with

κ= 1

∞−

1

p+1=1∈[0, 1], we conclude that there existα∈]0, 1− d

4p[andβ∈]0, 1[such that

E

•

1¯_AM

n(t)∩A¯Mn(s)

J₃1,5(t,s,x,y)

q˜

≤ C ndq”_|t₋s_|α+_|x₋ y_|β—q sup

r∈[0,T] Eh1A¯M

n(r)kXn(r,·)−X − n(r,·)k

q q

i

≤ C ndq2−nqσ”_|t₋s_|α+_|x₋ y_|β—q. (4.38)

From the above estimations, it follows that there exist ¯α ∈]0, min{1₂ − 3₄d_p,₂1(1− d₄),ρ₄}[and ¯β ∈

]0, min{2−3_pd, 2−d₂,2_d,ρ}[such that for(t,x),(s,y)_{∈ OT},

Eh1A¯M

n(t)∩A¯Mn(s)

J₃1(t,s,x,y)qi_≤C ndqh_|t₋s_|α¯+_|x₋y_|β¯iq. (4.39)

Now we turn to estimate the term J₃2(t,s,x,y). The procedure is similar to that of J₃1(t,s,x,y). Replace F(X_n(r,z)) by H(X_n(r,z)), F(X_n−(r,z)) by H(X_n−(r,z)), W(dz, dr) by W_n(dz, dr), and αn(r,z,Xn(r,z)) by βn(r,z,Xn(r,z)), respectively. Then there exist ¯α, ¯β presented in (4.39) such

that

E

h

1_A¯M

n(t)∩A¯Mn(s)

J₃2(t,s,x,y)qi _≤ C nq

h

|t−s|α¯+_|x−y|β¯iq. (4.40)

As for the termJ₃3(t,s,x,y), we have

E

h

1A¯M

n(t)∩A¯Mn(s)

J₃3(t,s,x,y)qi

= E

 1_A¯M

n(t)∩A¯Mn(s)

Z t

0

Z

D

[η(t,s,x,y)](r,z)H˙(X−_n(r,z))T_n3(r,z)W˙n(z,r)dzdr

q



= B1_n(t,s,x,y) +B2_n(t,s,x,y), (4.41)

where

B_n1(t,s,x,y) = E

–

1¯_AM

n(t)∩A¯Mn(s)

Z t

0

Z

D

×

Using the B-D-G inequality

B2_n(t,s,x,y)_≤E

By virtue of (4.7),

withδ′= ι∧α₂′. Finally we considerJ₃4(t,s,x,y) andJ₃5(t,s,x,y). From (A.11) and (A.12) with κ:= 1

∞−

1

p+1∈[0, 1]and (4.8), it follows that forα∈]0, 1− d

4p[andβ∈]0, 1[,

Eh1¯_AM

n(t)∩A¯Mn(s)

J₃4(t,s,x,y)qi

≤ C ndq212nq”|t−s|α+|x−y|β—q sup

r∈[0,T] Eh1A¯M

n(r)kT

4

n(r,·)k q p

i

≤ C n2dq2−12nq”|t−s|α+|x−y|β—q. (4.48)

As forJ₃5(t,s,x,y), using (A.11) and (A.12) withκ:= 1

∞−

2

p+1 :=1−

2

p∈[0, 1]and Lemma 4.2,

we get forα∈]0, 1−₂d_p[andβ∈]0, min{4−2_pd, 1}[,

Eh1A¯M

n(t)∩A¯Mn(s)

J₃5(t,s,x,y)qi

= E

 1A¯M

n(t)∩A¯Mn(s)

Z t

0

Z

D

[η(t,s,x,y)](r,z)ρ_n(r,z)_W˙_n(z,r)dzdr

q



≤ E

 1A¯M

n(t)∩A¯Mn(s)

Z t

0

Z

D

[η(t,s,x,y)](r,z)_|X_n(r,z)₋X_n−(r,z)_|2W˙_n(z,r)dzdr

q



≤ C ndq22( 1 2−

d

4)nq”|t−s|α+|x−y|β—q.

When d = 3, we can obtain a more precise estimate than the cases of d = 1, 2, which will be

concluded in Lemma 6.1 of Section 6. Thus we complete the proof of the lemma. ƒ

5

The proof of (C1)

The condition (C1) presented by Section 3 shall be verified in this section. By Lemma 4.1, to check (C1), we only need to prove the right hand side of (4.1) approaches 0, whenn_{→ ∞}. Note that for each fixedn∈N,

  

1_△j,k(x,t)

Æ

_△j,k ; j=0, 1, . . . , 2

n

−1, k∈Id_n

  

forms a CONS of L2([0,T]_×D). Letπn be the orthogonal projection of above basis and for any

mappingg:R_→R, define

τng(s) = g

€

(s+T2−n)_∧TŠ, s_∈[0,T].

Then for eacht_∈[0,T]andF-predictable process(ψ(t,x);x _∈D)_0≤t≤T,

Z t

0

Z

D

ψ(s,y)W_n(dy, ds) =

Z T

0

Z

D

πn

”

τn

€

RecallΛ_n(t,x)defined by (3.2) and we have

Then by the Hölder inequality and Burkhölder’s inequality, and note thatπn is an orthogonal

−1_[0,t](s)G_t−s(_•,y)H(X−_n(s,y))

Take the boundedness of the mappingHinto account, from (A.8) and (A.9) in Lemma A.2, it follows that forγ′′<1₋d₄,

sup

t∈[0,T]

”

|Λ˜1,1,1_n (t)_|+_|Λ˜1,1,3_n (t)_|—_≤C2−12γ′′nq. (5.1)

On the other hand, applying (A.16) withκ= 2

q−

2

q+1=1, the H¨older inequality, Lemma 4.2, and

Lemma 6.1 (in Section 6) to conclude that

+C sup

Then the B-D-G inequality yields that forq≥p,

×

Then from Lemma 4.2, and Lemma 6.1 (in Section 6), it follows that

and hence forq_≥p>3₂d,

sup

t∈[0,T] Eh1A¯M

n(t)

_Λ˜1

n(t,·)

q

p

i

→0, n_{→ ∞}. (5.7)

Finally, we turn to the estimation of the term ˜Λ2_n(t,x). In light of the B-D-G inequality and (A.16) withκ= 2_p−2_p+1=1,

Eh1A¯M n(t)

_Λ˜2

n(t,·)

q

p

i

≤ CE

1A¯Mn(t)

–Z t

0

Z

D

G2_t−s(_·,y)_|H(X_n(s,y))₋H(X_n−(s,y))_|2dyds

™

q

2

p

2

≤ CE

–Z t

0

(t−s)−d4kX_n(s,·)−X−

n(s,·)k p qds

™

≤ C2−ιnq. (5.8)

Observe that

˜

Λ3_n(t,x) =J3(t, 0,x, 0), (5.9)

for the termJ3(t,s,x,y)defined in Lemma 4.5. Then there exists aλ >0 such that forq≥p>3₂d,

sup

t∈[0,T] E

h

1A¯M n(t)

Λ˜3_n(t,·)q_pi_≤C2−λnq. (5.10)

Thus we prove that the condition (C1) holds. ƒ

6

The proof of (C2)

The aim of this section is to check the validity of the condition (C2) presented in Section 3. Note that for alls<t _∈[0,T], we have forq_≥p> 3₂d,

Eh1A¯M

n(t)∩A¯Mn(s)

Y_n(t,_·)₋Y_n(s,_·)q_pi

≤ CEh1¯_AM

n(t)∩A¯Mn(s)

X_n(t,_·)₋X_n(s,_·)q_pi+CEh1A¯M

n(t)∩A¯Mn(s)kX(t,·)−X(s,·)k q p

i

.

Observe the forms of Equations (2.2) and (2.3),X is a particular case ofXn. Hence in order to prove

that (C2) holds, it suffices to check that for all 0_≤s< t _≤ T, there existq _≥p andθ >qα¯( ¯α is presented in Theorem 1.1) such that for eachn_∈N,

(C2)′ E

h

1_A¯M

n(t)∩A¯Mn(s)

Xn(t,·)−Xn(s,·)

q

p

i

≤C|t−s|1+θ.

From (2.2), it follows that for(s,y),(t,x)_{∈ OT},