On the small time asymptotics of the two-dimensional stochastic Navier-Stokes equations

www.imstat.org/aihp 2009, Vol. 45, No. 4, 1002–1019

DOI: 10.1214/08-AIHP192

© Association des Publications de l’Institut Henri Poincaré, 2009

On the small time asymptotics of the two-dimensional stochastic Navier–Stokes equations

Tiange Xu and Tusheng Zhang

Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England, UK. E-mail: tzhang@maths.man.ac.uk Received 1 February 2008; revised 27 August 2008; accepted 10 September 2008

Abstract. In this paper, we establish a small time large deviation principle (small time asymptotics) for the two-dimensional stochastic Navier–Stokes equations driven by multiplicative noise, which not only involves the study of the small noise, but also the investigation of the effect of the small, but highly nonlinear, unbounded drifts.

Résumé. Dans cet article, nous établissons un principe de grandes déviations en temps petit pour l’équation de Navier–Stokes bi-dimensionnelle stochastique conduite par un bruit multiplicatif. Celui-ci nécessite non seulement l’étude d’un bruit faible, mais aussi la compréhension des effets de dérives petites mais non bornées et non linéaires.

MSC:Primary 60H15; secondary 60F10; 35Q30

Keywords:stochastic Navier–Stokes equation; small time asymptotics; large deviation principle

1. Introduction

It is well known that the two-dimensional stochastic Navier–Stokes equation with Dirichlet boundary condition de- scribes the time evolution of an incompressible fluid and is given by

⎧⎪

⎨

⎪⎩

du−νudt+(u· ∇)udt+ ∇pdt=gdt+σ (t, u)dW (t ),

(divu)(t, x)=0 for x∈D, t >0,

u(t, x)=0 for x∈∂D, t >0,

u(0, x)=u0(x) for x∈D,

whereDis a bounded domain inR²with smooth boundary∂D,u(t, x)∈R²denotes the velocity field at timet and positionx,p(t, x)denotes the pressure field,ν >0 is the viscosity andW is a Brownian motion on a Hilbert space.

Moreover,σ (t, u)dW is the random force field acting on the fluid andgis the deterministic part of the force.

To formulate the stochastic Navier–Stokes equations, we introduce the following standard spaces:

V = v∈H₀¹

D;R²

:∇ ·v=0 a.e., inD , with the norm

vV :=

D

|∇v|²dx 1/2

= v

and denote by((·,·))the inner product ofV.H is the closure ofV in theL²-norm

|v|H:=

D

|v|²dx 1/2

= |v|.

The inner product onHwill be denoted by(·,·).

Define the operatorA(Stokes operator) inHby the formula Au= −νP_Hu, ∀u∈H²

D;R²

∩V ,

where the linear operatorP_H(Helmhotz–Hodge projection) is the projection operator fromL²toH, and the nonlinear operatorB

B(u, v)=P_H

(u· ∇)v ,

with the notationB(u)=B(u, u). Obviously the domain ofBrequires that(u· ∇v)belongs to the spaceL². By applying the operatorP_H to each term of the above stochastic Navier–Stokes equation (SNSE), we can rewrite the SNSE in the following abstract form:

du(t )+Au(t )dt+B(u(t ))dt=f (t )dt+σ t, u(t )

dW (t ) inL²(0, T;V) (1.1)

with the initial condition

u(0)=u₀ inH, (1.2)

whereW (t )is a Brownian motion taking values in a Hilbert spaceU, andVis the dual ofV.

There exists a great amount of literature on the stochastic Navier–Stokes equation. Let us mention some of them.

A good reference for stochastic Navier–Stokes equations driven by additive noise is the book [4] and the references therein. The existence and uniqueness of solutions for stochastic 2-D Navier–Stokes equations with multiplicative noise were obtained in [8], [13] and [14]. The ergodic properties and invariant measures of the stochastic 2-D Navier–

Stokes equations were studied in [9] and [11]. The small noise large deviation of the stochastic 2-D Navier–Stokes equations was established in [14] and the large deviation of occupation measures was considered in [10].

The purpose of this paper is to study the small time asymptotics (large deviations) of the two-dimensional sto- chastic Navier–Stokes equations driven by multiplicative noise onC([0,1];H ). After the existence, uniqueness and continuity of the solution of SNSE are established, it is still generally difficult to quantify the behavior of the solution at a position x and a positive timet. It might be relatively simple to estimate the limiting behavior of the solution in time interval[0, t]ast goes to zero, which describes the behavior of the velocity of the fluid at a given point in space when the time is very small. This is one of the motivations for studying the small time asymptotics. Another motivation will be to get the following Varadhan identity through the small time asymptotics:

lim

t→02tlogP

u(0)∈B, u(t )∈C

= −d²(B, C), (1.3)

whered is an appropriate Riemannian distance associated with the diffusion generated by the solution of the SNSE.

This will be a topic of future study.

Apart from the above motivations, the small time asymptotics is also theoretically interesting, since the study involves the investigation of the small noise and the effect of the small, but highly nonlinear drift. We like to mention that the study of the small time asymptotics (large deviations) of finite dimensional diffusion processes was initiated by Varadhan in the influential work [16]. The small time asymptotics of infinite dimensional diffusion processes were studied in [1,2,7,12] and [17].

To establish the small time large deviation for the stochastic Navier–Stokes equation, as one expects, the main difficulty lies in dealing with the nonlinear term B(u)=P_H((u· ∇)u) and the unbounded term Au= −νP_Hu, which barely belong toV. To controlB(u), our idea is to show that the probability that the solution stays outside an energy ball is exponentially small so that we can restrict the solution in a sufficiently large energy ball. We hope

that this method used for treating the non-linear term could be useful in some other study of stochastic Navier–Stokes equations. Another key step of obtaining the small time large deviation is to prove that the law of the solution of the Navier–Stokes equationu(εt )is exponentially equivalent to the law of the solutions of the equation:

v^ε(t )=x+√ ε

_t

0

σ

εs, v^ε(s)

dW (s). (1.4)

This is done through several approximations. A remarkable martingale inequality proved by Barlow, Davis and Yor in [5] and [3], plays an important role throughout the paper. We point out that this martingale inequality is more precise than most of the B–D–G’s inequalities stated in the literature because the constant on the right isp^1/2(see (3.11)), but notp.

The rest of the paper is organized as follows. In Section 2, we collect some preliminaries which are frequently used in the sequel. Section 3 is the main part of the paper, where we proved the small time large deviations for stochastic Navier Stokes equation. In Section 4, we further relax the conditions on the diffusion coefficientσ (·).

2. Preliminaries

IdentifyingHwith its dualH, we consider Eq. (1.1) in the framework of Gelfrand triple:

V ⊂H∼=H⊂V.

In this way, we may considerAas a bounded operator fromV intoV. Moreover, we also denote by·,·, the duality betweenV andV. Hence, foru=(u_i)∈V,w=(w_i)∈V, we have

Au, w =ν

i,j

D

∂_iu_j∂_iw_jdx=ν (u, w)

. (2.1)

Introduce a trilinear form onH×H×Hby setting b(u, v, w)=

2 i,j

D

u_i∂_iv_jw_jdx, (2.2)

whenever the integral in (2.2) makes sense. In particular, ifu, v, w∈V, then B(u, v), w

=

(u· ∇v), w

= 2

i,j

D

u_i∂_iv_jw_jdx=b(u, v, w).

By the integration by parts,

b(u, v, w)= −b(u, w, v), (2.3)

therefore

b(u, v, v)=0, ∀u, v∈V . (2.4)

There are some well-known estimates forb(see [15], for example), which will be required in the rest of this paper and we list them here. Throughout the paper, we denote various generic positive constants by the same letterc. We have

b(u, v, w)≤cu · v · w|, (2.5)

b(u, v, w)≤c|u| · v · |Aw|, (2.6)

b(u, v, w)≤cu · |v| · |Aw|, (2.7)

b(u, v, w)≤2u^1/2· |u|^1/2· w^1/2· |w|^1/2· v (2.8)

for suitableu, v, w.Moreover, combining (2.3) and (2.8), we obtain a useful estimate as follows:

|Bu|V= sup

v≤1

b(u, u, v)= sup

v≤1

b(u, v, u)≤2u · |u|. (2.9)

3. Small time asymptotics

First, we introduce the precise assumptions on σ. Let(Ω,F, P ) be a complete probability space with a filtration Ft, t≥0 that satisfies the usual conditions. LetW (·)be aH-valued Brownian motion on(Ω,F, P )with the covari- ance operatorQ, which is a positive, symmetric, trace class operator onH. LetL_Qdenote the class of linear operators T such thatT Q^1/2is a Hilbert–Schmidt operator fromH toH.LQis endowed with the norm|T|²_LQ=Tr(T QT^∗), where Tr(T QT^∗)denotes the trace of operatorT QT^∗. LetL^V_Qdenote the class of linear operatorsT˜ such thatT Q˜ ^1/2 is a Hilbert–Schmidt operator fromH toV, endowed with the norm| ˜T|²_LV

Q=Tr(T Q˜ T˜^∗). Introduce:

(A.1) E|u0|⁴<+∞, f∈L⁴(0, T;V).

(A.2) There exists a constantLsuch that|σ (t, u)|²_LQ≤L(1+ |u|²),for allt∈(0, T ),and allu∈H. (A.3) There exists a constantLˆ such that for|σ (t, u)|²_LV

Q

≤ ˆL(1+ u²),for allt∈(0, T ),and allu∈V. (A.4) There exists a constantKsuch that|σ (t, u)−σ (t, v)|²_LQ≤K|u−v|²,for allt∈(0, T ),and allu, v∈H. (A.5) There exists a constantKˆ such that|σ (t, u)−σ (t, v)|²_LV

Q

≤ ˆKu−v²,for allt∈(0, T )and allu, v∈V. It is known that (see, for example, [14]), under the assumptions (A.1), (A.2) and (A.4), the SNSE

du(t )+Au(t )dt+B u(t )

dt=f (t )dt+σ t, u(t )

dW (t ),

u(0)=x∈H (3.1)

has a unique strong solutionu∈L²(Ω;C([0, T];H ))∩L²(Ω× [0, T];V ), i.e., u(t )=x−

_t

0

Au(s)ds− _t

0

B u(s)

ds+ _t

0

f (s)ds+ _t

0

σ s, u(s)

dW (s). (3.2)

Let ε >0, by the scaling property of the Brownian motion, it is easy to see thatu(εt ) coincides in law with the solution of the following equation:

u^ε(t )=x−ε _t

0

Au^ε(s)ds−ε _t

0

B u^ε(s)

ds+ε _t

0

f (εs)ds+√ ε

_t

0

σ

εs, u^ε(s)

dW (s). (3.3)

Letμ^ε_xbe the law ofu^ε(·)onC([0,1];H ).Define a functionalI (g)onC([0,1];H )by I (g)= inf

h∈Γg

1 2

₁

0

h(t )˙ ²

H0dt

,

whereH₀:=Q^1/2Hendowed with the norm|h|²_H0= |Q^−1/2h|²(h∈H₀), and Γ_g=

h∈C

[0, T];H

: h(·)is absolutely continuous and such that g(t )=x+

_t

0

σ

s, g(s)h(s)˙ ds,0≤t≤1

.

Theorem 3.1. μ^ε_xsatisfies a large deviation principle with the rate functionI (·),that is, (i) For any closed subsetF ⊂C([0,1];H ),

lim sup

ε→0,xn→x

εlogμ^ε_x

n(F )≤ −inf

g∈F

I (g) . (ii) For any open subsetG⊂C([0,1];H ),

lim inf

ε→0,x_n→xεlogμ^ε_x

n(G)≥ −inf

g∈G

I (g) .

Proof. Letv^ε(·)be the solution of the stochastic equation v^ε(t )=x+√

ε _t

0

σ

εs, v^ε(s)

dW (s), (3.4)

andν^ε be the law ofv^ε(·)on the C([0,1];H ). Then by [4], we know that ν^ε satisfies a large deviation principle with the rate functionI (·). Our main task is to show that two families of the probability measuresμ^ε andν^ε are exponentially equivalent, that is, for anyδ >0,

εlim→0εlogP

sup

0≤t≤1

u^ε(t )−v^ε(t )²> δ

= −∞. (3.5)

Then Theorem 3.1 follows from (3.5) and Theorem 4.2.13 in [6] forx_n=x.Slight modifications of the proof yields

the general case.

Because of the non-linear formB(·,·,·), and the unbounded operatorA, the proof of (3.5) is quite involved. We split it into several lemmas. The following result is an estimate of the probability that the solution of (3.3) leaves an energy ball.

Lemma 3.1.

Mlim→∞ sup

0<ε≤1

εlogPu^εH

V(1)2

> M

= −∞, (3.6)

where(|u^ε|^H_V(1))²=sup_0≤t≤1|u^ε(t )|²+εν₁

0u^ε(t )²dt. Proof. SinceB(u^ε(s)), u^ε(s) =0, applying Itô’s formula , we get

u^ε(t )²= |x|²−2ε _t

0

u^ε(s), Au^ε(s)ds+2ε _t

0

u^ε(s), f (εs) ds +2√

ε _t

0

u^ε(s), σ

εs, u^ε(s) dW (s)

+ε _t

0

σ

εs, u^ε(s)²

LQds, that is,

u^ε(t )²+2εν _t

0

u^ε(s)²ds= |x|²+2ε _t

0

u^ε(s), f (εs)

ds+2√ ε

_t

0

u^ε(s), σ

εs, u^ε(s) dW (s) +ε

_t

0

σ

εs, u^ε(s)²

LQds

= |x|²+l1+l2+l3. (3.7)

Forl₁, we have

|l1| ≤2ε _t

0

u^ε(s)·f (εs)

Vds≤εν _t

0

u^ε(s)²ds+1

ν|f|²_L2(0,εt;V). (3.8)

In view of (A.2),

|l3| ≤ε·L _t

0

1+u^ε(s)²

ds. (3.9)

Putting(3.7), (3.8)and(3.9)together, it follows that u^ε(t )²+εν

_t

0

u^ε2ds≤

|x|²+εLt+1

ν|f|²_L2(0, εt;V)

+εL

_t

0

u^ε(s)²ds +2√

ε _t

0

u^ε(s), σ

εs, u^ε(s)dW (s). Therefore,

u^εH

V(T )2

≤2

|x|²+εLT+1

ν|f|²_L2(0,εT;V)

+2εL

_T

0

u^ε(s)^H

V(s)2

ds +4√

ε sup

0≤t≤T

_t

0

u^ε(s), σ

εs, u^ε(s)dW (s). Hence, forp≥2, we have,

Eu^εH

V(T )2p1/p

≤2

|x|²+εLT+1

ν|f|²_L2(0,εT;V)

+2εL

E

_T

0

u^εH

V(s)2

ds _p1/p

+4√ ε

E

sup

0≤t≤T

_t

0

(u^ε(s), σ

εs, u^ε(s)

dW (s)) ^p1/p

. (3.10)

To estimate the stochastic integral term, we will use the following remarkable result from [3] and [5] that there exists a universal constantcsuch that, for anyp≥2 and for any continuous martingale(Mt)withM0=0, one has

M_t^∗

p≤cp^1/2M^1/2t

p, (3.11)

whereM_t^∗=sup_0≤s≤t|M_s|and · pstands for theL^P-norm.

Using this result, we have 4√

ε

E

sup

0≤t≤T

_t

0

u^ε(s), σ

εs, u^ε(s)

dW (s)^p1/p

≤4c√ pε

E

_T

0

u^ε(s)²·σ

εs, u^ε(s)²

LQds

p/21/p

≤4c√ pε

E

_T

0

u^ε(s)²

1+u^ε(s)² ds

p/21/p

≤4c√ pε

E

T 0

1+u^ε(s)²2

ds

p/22/p1/2

≤4c√ pε

E

_T

0

1+u^ε(s)⁴ ds

p/22/p1/2

≤4c√ pε

_T

0

1+

Eu^ε(s)^2p2/p

ds 1/2

, (3.12)

wherecis a constant which may change from line to line. On the other hand, 2εL

E

_T

0

u^εH

V(s)2

ds p1/p

≤2εL _T

0

Eu^εH

V(s)2p1/p

ds. (3.13)

Combining(3.10), (3.12)and(3.13), we arrive at Eu^εH

V(T )2p2/p

≤8

|x|²+εLT +1

ν|f|²_L2(0, εT;V)

2

+8ε²L²T _T

0

Eu^εH

V(s)2p2/p

ds +32c²pεT +32c²pε

_T

0

Eu^εH

V(s)2p2/p

ds. (3.14)

Applying the Gronwall’s inequality, we obtain Eu^εH

V(1)2p2/p

≤

8

|x|²+εL+1

ν|f|²_L2(0,ε;V)

2

+32c²pε

·exp

8ε²L²+32c²pε

. (3.15)

SinceP ((|u^ε|^H_V(1))²> M)≤M^−pE(|u^ε|^H_V(1))^2p, letp=¹_ε in(3.15)to get εlogPu^εH

V(1)2

> M

≤ −logM+logEu^εH

V(1)2p1/p

≤ −logM+log

8

|x|²+εL+1

ν|f|²_L2(0, ε;V)

2

+32c²

+4ε²L²+16c². Therefore,

sup

0<ε≤1

εlogPu^εH

V(1)2

> M

≤ −logM+log

8

|x|²+L+1

ν|f|²_L2(0,1;V)

2

+32c²

+16c²+4L².

LettingM→ ∞on both side of the above inequality, we finish the proof.

SinceV is dense inH, there exists a sequence{x_n}^∞_n=1⊂V such that

n→+∞lim |x_n−x| =0.

Letu^ε_n(·)be the solution of (3.3) with the initial valuex_n. From the proof of Lemma 3.1, it follows that

M→+∞lim sup

n

sup

0<ε≤1

εlogPu^ε_n^H

V(1)2

> M

= −∞. (3.16)

Letv^ε_n(·)be the solution of(3.4)with the initial valuex_n, and we have the following result.

Lemma 3.2. For anyn∈Z⁺,

Mlim→∞ sup

0<ε≤1

εlogP

sup

0≤t≤1

v_n^ε(t )²> M

= −∞.

Proof. Applying Itô’s formula tov_n^ε(t )², one obtains v_n^ε(t )²= x_n²+2√

ε _t

0

v^ε_n(s), σ

εs, v_n^ε(s)

dW (s) +ε

_t

0

σ

εs, v^ε_n(s)²

L^V_Qds.

By (A.3) and inequality(3.11), we have

E

sup

0≤t≤r

v_n^ε(t )^2p2/p

≤2x_n⁴+8cεp

E r

0

v^ε_n(s)²σ

εs, v^ε_n(s)²

L^V_Qds

p/22/p

+4ε²Lˆ²r

r+ _r

0

E

sup

0≤l≤s

v^ε_n(l)^2p2/p

ds

≤2xn⁴+16cεpLˆ

r+ _r

0

E

sup

0≤l≤s

v^ε_n(l)^2p2/p

ds

+4ε²Lˆ²r

r+ _r

0

E

sup

0≤l≤s

v^ε_n(l)^2p2/p

ds

. (3.17)

By Gronwall’s inequality,

E

sup

0≤t≤1

v^ε_n(t )^2p2/p

≤

2xn⁴+16cεpLˆ+4ε²Lˆ²

e^{16cεpLˆ+4ε2Lˆ2}. (3.18)

The rest of the proof is the same as that of Lemma 3.1.

Lemma 3.3. For anyδ >0,

n→+∞lim sup

0<ε≤1

εlogP

sup

0≤t≤1

u^ε(t )−u^ε_n(t )²> δ

= −∞. (3.19)

Proof. ForM >0, define a stopping time τε,M=inf

t: εν

_t

0

u^ε(r)²dr > M, oru^ε(t )²> M

. Clearly,

P

sup

0≤t≤1

u^ε(t )−u^ε_n(t )²> δ,u^εH

V(1)2

≤M ≤P

sup

0≤t≤1

u^ε(t )−u^ε_n(t²> δ, τε,M≥1

≤P

sup

0≤t≤1∧τ_ε,M

u^ε(t )−u^ε_n(t )²> δ

. (3.20)

Letkbe a positive constant. Applying Itô’s formula to e^−kε

t∧τε,M

0 u^ε(s)²ds|u^ε(t∧τ_ε,M)−u^ε_n(t∧τ_ε,M)|², we get e^−kε

t∧τε,M

0 u^ε(s)²dsu^ε(t∧τ_ε,M)−u^ε_n(t∧τ_ε,M)²+2εν _t∧τε,M

0

e^−kε

_s

0u^ε(r)²dru^ε(s)−u^ε_n(s)²ds

= |x−xn|²−kε _t∧τε,M

0

e^{−kε0suε(r)2dr}u^ε(s)²u^ε(s)−u^ε_n(s)²ds

−2ε _t∧τε,M

0

e^−kε

s

0u^ε(r)²dr b

u^ε(s), u^ε(s), u^ε(s)−u^ε_n(s)

−b

u^ε_n(s), u^ε_n(s), u^ε(s)−u^ε_n(s) ds

+ε _t∧τε,M

0

e^{−kε0suε(r)2dr}σ

εs, u^ε(s)

−σ

εs, u^ε_n(s)²

LQds +2√

ε _t∧τε,M

0

e^−kε

s

0u^ε(r)²dr

u^ε(s)−u^ε_n(s), σ

εs, u^ε(s)

−σ

εs, u^ε_n(s) dW (s)

. (3.21)

Notice that by (2.4), we have b

u^ε_n(t ), u^ε_n(t ), u^ε(t )−u^ε_n(t )

=b

u^ε_n(t ), u^ε(t ), u^ε(t )−u^ε_n(t ) , hence by (2.9),

b

u^ε(t ), u^ε(t ), u^ε(t )−u^ε_n(t )

−b

u^ε_n(t ), u^ε_n(t ), u^ε(t )−u^ε_n(t )

=b

u^ε(t )−u^ε_n(t ), u^ε(t ), u^ε(t )−u^ε_n(t )

≤2u^ε(t )−u^ε_n(t )·u^ε(t )·u^ε(t )−u^ε_n(t ). Therefore, by(3.21),

e^−kε

t∧τε,M

0 u^ε(s)²dsu^ε(t∧τ_ε,M)−u^ε_n(t∧τ_ε,M)²+2εν _t∧τε,M

0

e^−kε

s

0u^ε(r)²dru^ε(s)−u^ε_n(s)²ds

≤ |x−x_n|²−kε _t∧τε,M

0

e^−kε

s

0u^ε(r)²dru^ε(s)²u^ε(s)−u^ε_n(s)²ds +4ε

_t∧τε,M

0

e^−kε

_s

0u^ε(r)²dru^ε(s)−u^ε_n(s)·u^ε(s)·u^ε(s)−u^ε_n(s)ds +εK

_t∧τε,M

0

e^{−kε0suε(r)2dr}u^ε(s)−u^ε_n(s)²ds +2√

ε _t∧τε,M

0

e^−kε

s

0u^ε(r)²dr

u^ε(s)−u^ε_n(s), σ

εs, u^ε(s)

−σ

εs, u^ε_n(s)dW (s)

≤ |x−x_n|²−kε _t∧τε,M

0

e^−kε

s

0u^ε(r)²dru^ε(s)²u^ε(s)−u^ε_n(s)²ds +εν

_t∧τε,M

0

e^−kε

_s

0u^ε(r)²dru^ε(s)−u^ε_n(s)²ds +4ε

ν

_t∧τε,M

0

e^{−kε0suε(r)2dr}u^ε(s)−u^ε_n(s)²·u^ε(s)²ds +εK

_t∧τε,M

0

e^−kε

s

0u^ε(r)²dru^ε(s)−u^ε_n(s)²ds +2√

ε _t∧τε,M

0

e^−kε

_s

0u^ε(r)²dr

u^ε(s)−u^ε_n(s), σ

εs, u^ε(s)

−σ

εs, u^ε_n(s)dW (s). (3.22) Choosingk >⁴_ν and using(3.11), we have,

E

sup

0≤s≤t∧τε,M

e^{−k0suε(r)2dr}u^ε(s)−u^ε_n(s)^2p2/p

≤2|x−xn|⁴+2ε²K² _t

0

E

sup

0≤r≤s∧τε,M

e^{−kε0ruε(l)2dl}u^ε(r)−u^ε_n(r)²p2/p

ds +8cεpK²

t 0

E

sup

0≤r≤s∧τε,M

e^−2kε

r

0u^ε(l)²dlu^ε(r)−u^ε_n(r)⁴p/22/p

ds

≤2|x−xn|⁴+2ε²K² _t

0

E

sup

0≤r≤s∧τε,M

e^{−kε0ruε(l)2dl}u^ε(r)−u^ε_n(r)²p2/p

ds +8cεpK²

_t

0

E

sup

0≤r≤s∧τε,M

e^{−kε0ruε(l)2dl}u^ε(r)−u^ε_n(r)²p2/p

ds. (3.23)

Applying Gronwall’s inequality, one obtains,

E

sup

0≤t≤1∧τ_ε,M

e^−kε

_t

0u^ε(s)²dsu^ε(t )−u^ε_n(t )²_p2/p

≤2|x−xn|⁴e^{2ε2K2+8cεpK2}. (3.24)

Hence,

E

sup

0≤t≤1∧τ_ε,M

u^ε(t )−u^ε_n(t )²p2/p

≤ E

sup

0≤t≤1∧τε,M

e^{−kε0tuε(s)2ds}u^ε(t )−u^ε_n(t )²p

e^kpε

1∧τε,M

0 u^ε(s)²ds2/p

≤e^2kM

E

sup

0≤t≤1∧τ_ε,M

e^−kε

t

0u^ε(s)²dsu^ε(t )−u^ε_n(t )²p2/p

≤2e^2kM|x−x_n|⁴e^{2ε2K2+8cεpK2}. (3.25)

FixM, and takep=2/εto get sup

0<ε≤1

εlogP

sup

0≤t≤1∧τε,M

u^ε(t )−u^ε_n(t )²> δ

≤ sup

0≤ε≤1

εlogE[sup_{0≤t≤1∧τε,M}|u^ε(t )−u^ε_n(t )|^2p] δ^p

≤2kM+2K²+16cK²−2 logδ+log 2|x−x_n|⁴

→ −∞, asn→ +∞. (3.26)

By Lemma 3.1, for anyR >0, there exists a constantMsuch that for anyε∈(0,1], the following inequality holds, Pu^εH

V(1)2

> M

≤e^−R/ε. (3.27)

For such aM, by(3.20), (3.26), there exits a positive integerN, such that for anyn≥N, sup

0<ε≤1

εlogP

sup

0≤t≤1

u^ε(t )−u^ε_n(t )²> δ,u^εH

V(1)2

≤M

≤ −R. (3.28)

Putting(3.27)and(3.28)together, one sees that there exists a positive integerN, such that for anyn≥N, ε∈(0,1] P

sup

0≤t≤1

u^ε(t )−u^ε_n(t )²> δ

≤2e^−R/ε. (3.29)

SinceRis arbitrary, the lemma follows.

We have the following similar result forv^ε(·)andv_n^ε(·).