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J. Differential Equations 258 (2015) 3363–3390

www.elsevier.com/locate/jde

A moderate deviation principle for 2-D stochastic

Navier–Stokes equations

Ran Wang a, Jianliang Zhai a,∗, Tusheng Zhang b,a

a School of Mathematical Sciences, University of Science and Technology of China, Wu Wen Tsun Key Laboratory of Mathematics, Chinese Academy of Science, Hefei, 230026, China

b School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

Received 27 January 2014; revised 6 January 2015

Available online 22 January 2015

Abstract

In this paper, we prove a central limit theorem and establish a moderate deviation principle for two-dimensional stochastic Navier–Stokes equations with multiplicative noise. The weak convergence method plays an important role.© 2015 Elsevier Inc. All rights reserved.

MSC: primary 60H15; secondary 60F05, 60F10

Keywords: Stochastic Navier–Stokes equations; Central limit theorem; Moderate deviation principle

1. Introduction

Consider the two-dimensional stochastic Navier–Stokes equation with Dirichlet boundary condition, which describes the time evolution of an incompressible fluid,

∂uε(t)

∂t− �uε(t) + (

uε(t) · ∇)uε(t) + ∇p(t, x) = f (t) + √

εσ(t, uε(t)

)dW(t)

dt, (1.1)

* Corresponding author.E-mail addresses: wangran@ustc.edu.cn (R. Wang), zhaijl@ustc.edu.cn (J.L. Zhai),

Tusheng.Zhang@manchester.ac.uk (T.S. Zhang).

http://dx.doi.org/10.1016/j.jde.2015.01.0080022-0396/© 2015 Elsevier Inc. All rights reserved.

3364 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

with the conditions ⎧⎨⎩(∇ · uε

)(t, x) = 0, for x ∈ D, t > 0,

uε(t, x) = 0, for x ∈ ∂D, t ≥ 0,

uε(0, x) = u0(x), for x ∈ D,

(1.2)

where D is a bounded open domain of R2 with regular boundary ∂D, uε(t, x) ∈ R2 denotes the

velocity field at time t and position x, p(t, x) denotes the pressure field, f is a deterministic external force, and W(·) is a Wiener process.

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

V = {v ∈ H 1

0

(D;R2) : ∇ · v = 0, a.e. in D

},

with the norm

‖v‖V :=(∫

D

|∇v|2dx

) 12 = ‖v‖,

and let H be the closure of V in the L2-norm

|v|H :=(∫

D

|v|2dx

) 12 = |v|.

Define the operator A (Stokes operator) in H by the formula

Au := −PH �u, ∀u ∈ H 2(D;R2) ∩ V,

where the linear operator PH (Helmhotz–Hodge projection) is the projection operator from L2(D; R2) to H , and define the nonlinear operator B by

B(u, v) := PH

((u · ∇)v

),

with the notation B(u) := B(u, u) for short.By applying the operator PH to each term of (1.1), we can rewrite it in the following abstract

form:

duε(t) + Auε(t)dt + B(uε(t)

)dt = f (t)dt + √

εσ(t, uε(t)

)dW(t), (1.3)

with the initial condition uε(0) = x for some fixed point x in H .As the parameter ε tends to zero, the solution uε of (1.3) will tend to the solution of the

following deterministic Navier–Stokes equation

du0(t) + Au0(t)dt + B(u0(t)

)dt = f (t)dt, with u0(0) = x. (1.4)

R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390 3365

In this paper, we shall investigate deviations of uε from the deterministic solution u0, as εdecreases to 0, that is, the asymptotic behavior of the trajectory,

Y ε(t) = 1√ελ(ε)

(uε − u0)(t), t ∈ [0, T ],

where λ(ε) is some deviation scale which strongly influences the asymptotic behavior of Y ε.

(1) The case λ(ε) = 1/√

ε provides some large deviations estimates, which have been exten-sively studied in recent years.

(2) If λ(ε) is identically equal to 1, we are in the domain of the central limit theorem (CLT for short). We will show that (uε − u0)/

√ε converges to a solution of a stochastic equation as ε

decreases to 0.(3) To fill in the gap between the CLT scale [λ(ε) = 1] and the large deviations scale [λ(ε) =

1/√

ε ], we will study the so-called moderate deviation principle (MDP for short, cf. [8]), that is when the deviation scale satisfies

λ(ε) → +∞,√

ελ(ε) → 0 as ε → 0. (1.5)

Throughout this paper, we assume that (1.5) is in place.

Large deviations for stochastic partial differential equations have been investigated in many papers, see [4,5,19], etc. Since the work of Bensoussan and Temam [2], stochastic Navier–Stokes equations have been intensively studied, see [7] for the equation with additive Gaussian noise. The existence and uniqueness of solutions for the 2-D stochastic Navier–Stokes equations with multiplicative Gaussian noise were obtained in [11,20]. The ergodic properties and invariant measures of the 2-D stochastic Navier–Stokes equations were studied in [10] and [14]. Wentzell–Freidlin type large deviation results for the two-dimensional stochastic Navier–Stokes equations with Gaussian noise have been established in [20], and the case of Lévy noise has been estab-lished in [26] and [27].

Like the large deviations, the moderate deviation problems arise in the theory of statistical inference quite naturally. The estimates of moderate deviations can provide us with the rate of convergence and a useful method for constructing asymptotic confidence intervals, see [9,12,15,16] and references therein. Results on the MDP for processes with independent increments were obtained in De Acosta [1], Chen [6] and Ledoux [17]. The study of the MDP estimates for other processes has been carried out as well, e.g., Wu [25] for Markov processes, Guillin and Liptser [13] for diffusion processes, Wang and Zhang [24] for stochastic reaction–diffusion equations.

The organization of this paper is as follows. In Section 2, we shall give some preliminary results on stochastic two-dimensional Navier–Stokes equations. In Section 3, we establish the central limit theorem. Section 4 is devoted to establishing the moderate deviation principle.

Throughout this paper, cN, cf,T , · · · are positive constants depending on some parameters N, f, T , · · · , independent of ε, whose value may be different from line to line.

2. Stochastic Navier–Stokes equations

Let V ′ be the dual of V . Identifying H with its dual H ′, we have the dense, continuous embedding

3366 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

V ↪→ H ∼= H ′ ↪→ V ′.

In this way, we may consider A as a bounded operator from V to V ′. Moreover, we also denote by (·, ·), the duality between V and V ′. Hence, for u = (ui) ∈ V , w = (wi) ∈ V , we have

(Au,w) =2∑

i,j

∫D

∂iuj ∂iwjdx. (2.6)

Define b(·, ·, ·) : V × V × V → R by

b(u, v,w) =2∑

i,j

∫D

ui∂ivjwjdx. (2.7)

In particular, if u, v, w ∈ V , then

(B(u, v),w

) = ((u · ∇)v,w

) =2∑

i,j

∫D

ui∂ivjwjdx = b(u, v,w).

B(u) will be used to denote B(u, u). By integration by parts,

b(u, v,w) = −b(u,w,v), (2.8)

therefore

b(u, v, v) = 0, ∀u,v ∈ V. (2.9)

There are some well-known estimates for b (see [21] and [20] for example), which will be re-quired in the rest of this paper.∣∣b(u, v,w)

∣∣ ≤ 2‖u‖ 12 · |u| 1

2 · ‖v‖ 12 · |v| 1

2 · ‖w‖, (2.10)∣∣b(u,u, v)∣∣ ≤ 1

2‖u‖2 + c‖v‖4

L4 · |u|2, (2.11)∣∣(B(u) − B(v),u − v)∣∣ ≤ 1

2‖u − v‖2 + c|u − v|2 · ‖v‖4

L4, (2.12)

where

‖v‖4L4 ≤ ‖v‖2|v|2. (2.13)

By [23], B can be extended to a continuous operator

B : H × H → D(A−�

)(2.14)

for some � > 1.

R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390 3367

The covariance operator Q of the Wiener process W(·) is a positive symmetric, trace class operator on H . Let H0 = Q1/2H . Then H0 is a Hilbert space with the inner product

〈u,v〉0 = (Q−1/2u,Q−1/2v

) ∀u,v ∈ H0. (2.15)

Let | · |0 denote the norm in H0. Clearly, the embedding of H0 in H is Hilbert–Schmidt, since Q is a trace class operator. Let LQ(H0; H) denote the space of linear operators S such that SQ1/2 is a Hilbert–Schmidt operator from H to H . Define the norm on the space LQ(H0; H) by |S|LQ

= √tr(SQS∗).

Hypothesis. The noise coefficient σ : [0, T ] ×V → LQ(H0; H) and the force term f satisfy the following hypotheses:

(A.1) the function σ ∈ C([0, T ] × V ; LQ(H0; H));(A.2) there exists a positive constant K such that for all t ∈ [0, T ], u, v ∈ V ,∣∣σ(t, u)

∣∣2LQ

≤ K(1 + ‖u‖2) and

∣∣σ(t, u) − σ(t, v)∣∣2LQ

≤ K‖u − v‖2; (2.16)

(A.3) the force term f is in L4([0, T ]; V ′), that is

T∫0

∥∥f (s)∥∥4

V ′ds < ∞.

3. Central limit theorem

In this section, we will establish the central limit theorem.Let uε be the unique solution of Eq. (1.3) in L2(Ω; C([0, T ]; H)) ∩L2(Ω ×[0, T ]; V ), and u0

the unique solution of Eq. (1.4). The following estimates follow from Proposition 2.3 in Sritharan and Sundar [20].

Lemma 3.1. Assume that (A.1)–(A.3) hold. There exists a constant ε0 > 0 such that

(i) for all 0 < ε ≤ ε0,

E

(sup

0≤t≤T

∣∣uε(t)∣∣2 +

T∫0

∥∥uε(s)∥∥2

ds

)≤ cf,T ; (3.17)

(ii) for all 0 < ε ≤ ε0,

E

(sup

0≤t≤T

∣∣uε(t)∣∣4 +

T∫0

∣∣uε(s)∣∣2 · ∥∥uε(s)

∥∥2ds

)≤ cf,T , (3.18)

where cf,T is a constant independent of ε;

3368 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

(iii) particularly, it holds that

sup0≤t≤T

∣∣u0(t)∣∣2 +

T∫0

∥∥u0(s)∥∥2

ds ≤ cf,T ,

and

T∫0

∥∥u0(s)∥∥4

L4ds ≤ sup0≤t≤T

∣∣u0(t)∣∣2

T∫0

∥∥u0(s)∥∥2

ds ≤ cf,T .

The next result is concerned with the convergence of uε as ε → 0.

Proposition 3.1. Under the conditions (A.1)–(A.3), there exists a constant ε0 > 0 such that, for all 0 < ε ≤ ε0,

E

(sup

0≤t≤T

∣∣uε(t) − u0(t)∣∣2 +

T∫0

∥∥uε(s) − u0(s)∥∥2

ds

)≤ εcf,T ,K . (3.19)

Proof. Define τN = inf{t : |uε(t)|2 + ∫ t

0 ‖uε(s)‖2ds > N}. Applying Itô’s formula to |uε(t) −u0(t)|2, one obtains that

∣∣uε(t ∧ τN) − u0(t ∧ τN)∣∣2 + 2

t∧τN∫0

∥∥uε(s) − u0(s)∥∥2

ds

= −2

t∧τN∫0

(B

(uε(s)

) − B(u0(s)

), uε(s) − u0(s)

)ds

+ 2√

ε

t∧τN∫0

(σ(s, uε(s)

)dW(s), uε(s) − u0(s)

) + ε

t∧τN∫0

∥∥σ(s, uε(s)

)∥∥2LQ

ds. (3.20)

Taking the supremum up to time t ∧ τN in (3.20), and then taking the expectation, we have

E

(sup

0≤s≤t∧τN

[∣∣uε(s) − u0(s)∣∣2 + 2

s∧τN∫0

∥∥uε(l) − u0(l)∥∥2

dl

])

≤ 2E

( t∧τN∫ ∣∣(B(uε(s)

) − B(u0(s)

), uε(s) − u0(s)

)∣∣ds

)

0

R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390 3369

+ 2√

εE

(sup

0≤s≤t∧τN

∣∣∣∣∣s∫

0

(σ(l, uε(l)

)dW(l), uε(l) − u0(l)

)∣∣∣∣∣)

+ εE

( t∧τN∫0

∥∥σ(s, uε(s)

)∥∥2LQ

ds

)

=: I1(t) + I2(t) + I3(t). (3.21)

By (2.12), we have

∣∣I1(t)∣∣ ≤ 2E

t∧τN∫0

(1

2

∥∥uε(s) − u0(s)∥∥2 + c

∣∣uε(s) − u0(s)∣∣2 · ∥∥u0(s)

∥∥4L4

)ds

≤ E

( t∧τN∫0

∥∥uε(s) − u0(s)∥∥2

ds

)

+ 2c

t∫0

E

(sup

0≤l≤s∧τN

∣∣uε(l) − u0(l)∣∣2

)· ∥∥u0(s)

∥∥4L4ds. (3.22)

By the Burkholder–Davis–Gundy inequality, we have

∣∣I2(t)∣∣ = 2

√εE

(sup

0≤s≤t∧τN

∣∣∣∣∣s∫

0

(σ(l, uε(l)

)dW(l), uε(l) − u0(l)

)∣∣∣∣∣)

≤ 4√

εK E

( t∧τN∫0

(1 + ∥∥uε(s)

∥∥2)∣∣uε(s) − u0(s)∣∣2

ds

)1/2

≤ 4√

εK E

(sup

0≤s≤t∧τN

∣∣uε(s) − u0(s)∣∣ ·

( t∧τN∫0

(1 + ∥∥uε(s)

∥∥2)ds

)1/2)

≤ 1

2E

(sup

0≤s≤t∧τN

∣∣uε(s) − u0(s)∣∣2

)+ 8εKE

( t∧τN∫0

(1 + ∥∥uε(s)

∥∥2)ds

), (3.23)

where K is the constant appeared in (2.16). By (2.16), we obtain that

∣∣I3(t)∣∣ = εE

( t∧τN∫ ∥∥σ(s, uε(s)

)∥∥2LQ

ds

)≤ εKE

( t∧τN∫ (1 + ∥∥uε(s)

∥∥2)ds

). (3.24)

0 0

3370 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

Putting (3.21)–(3.24) together, we have

E

(sup

0≤s≤t∧τN

∣∣uε(s) − u0(s)∣∣2 +

t∧τN∫0

∥∥uε(s) − u0(s)∥∥2

ds

)

≤ c

t∫0

E

(sup

0≤l≤s∧τN

∣∣uε(l) − u0(l)∣∣2

)· ∥∥u0(s)

∥∥4L4ds + cεKE

t∧τN∫0

(1 + ∥∥uε(s)

∥∥2)ds. (3.25)

Applying Gronwall’s inequality to g(t) = E(sup0≤s≤t∧τN|uε(s) − u0(s)|2), and using Lem-

ma 3.1, we have

E

(sup

0≤s≤T ∧τN

∣∣uε(s) − u0(s)∣∣2

)≤ cεK

(E

T ∧τN∫0

(1 + ∥∥uε(s)

∥∥2)ds

)exp

{c

T∫0

∥∥u0(s)∥∥4

L4ds

}

≤ εcf,T ,K . (3.26)

Plugging (3.26) into the right hand side of (3.25), and letting N → 0, we further have

E

(sup

0≤t≤T

∣∣uε(t) − u0(t)∣∣2 +

T∫0

∥∥uε(s) − u0(s)∥∥2

ds

)≤ εcf,T ,K, (3.27)

which completes the proof. �Let V 0 be the solution of the following SPDE:

dV 0(t) + (AV 0(t) + B

(V 0(t), u0(t)

) + B(u0(t),V 0(t)

))dt = σ

(t, u0(t)

)dW(t), (3.28)

with initial value V 0(0) = 0. Using the classical Galerkin method, the existence and uniqueness of the solution for (3.28) can be proved similarly as for the case of 2-D stochastic Navier–Stokes equation. Furthermore, the solution has the following estimate

E

(sup

0≤t≤T

∣∣V 0(t)∣∣2

)+E

( T∫0

∥∥V 0(s)∥∥2

ds

)≤ cf,T .

Our first result is the following central limit theorem.

Theorem 3.2 (Central limit theorem). Under the conditions (A.1)–(A.3), (uε − u0)/√

ε con-verges to V 0 in the space L2(Ω; C([0, T ]; H)) ∩ L2(Ω × [0, T ]; V ), that is

limε→0

E

{sup

0≤t≤T

∣∣∣∣uε(t) − u0(t)√ε

− V 0(t)

∣∣∣∣2

+T∫

0

∥∥∥∥uε(s) − u0(s)√ε

− V 0(s)

∥∥∥∥2

ds

}= 0. (3.29)

R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390 3371

Proof. Let V ε(t) := (uε(t) − u0(t))/√

ε. Then V ε(0) = 0,

dV ε(t) + (AV ε(t) + B

(V ε(t), uε(t)

) + B(u0(t),V ε(t)

))dt = σ

(t, uε(t)

)dW(t),

and

d(V ε(t) − V 0(t)

) + A(V ε(t) − V 0(t)

)dt

+ (B

(V ε(t), uε(t)

) − B(V 0(t), u0(t)

))dt + B

(u0(t),V ε(t) − V 0(t)

)dt

= (σ(t, uε(t)

) − σ(t, u0(t)

))dW(t).

By Itô’s formula and (2.9), we have

d∣∣V ε(t) − V 0(t)

∣∣2 + 2∥∥V ε(t) − V 0(t)

∥∥2dt

= −2(B

(V ε(t), uε(t)

) − B(V 0(t), u0(t)

),V ε(t) − V 0(t)

)dt

+ 2((

σ(t, uε(t)

) − σ(t, u0(t)

))dW(t),V ε(t) − V 0(t)

)+ ∥∥σ

(t, uε(t)

) − σ(t, u0(t)

)∥∥2LQ

dt

= −2(B

(V ε(t) − V 0(t), u0(t)

) + B(V ε(t), uε(t) − u0(t)

),V ε(t) − V 0(t)

)dt

+ 2((

σ(t, uε(t)

) − σ(t, u0(t)

))dW(t),V ε(t) − V 0(t)

)+ ∥∥σ

(t, uε(t)

) − σ(t, u0(t)

)∥∥2LQ

dt.

Defining τN = inf{t : |V ε(t) − V 0(t)|2 + ∫ t

0 ‖V ε(s) − V 0(s)‖2ds > N}, we have

∣∣V ε(t ∧ τN) − V 0(t ∧ τN)∣∣2 + 2

t∧τN∫0

∥∥V ε(s) − V 0(s)∥∥2

ds

≤ 2

t∧τN∫0

∣∣(B(V ε(s) − V 0(s), u0(s)

),V ε(s) − V 0(s)

)∣∣ds

+ 2

t∧τN∫0

∣∣(B(V ε(s), uε(s) − u0(s)

),V ε(s) − V 0(s)

)∣∣ds

+ 2

∣∣∣∣∣t∧τN∫0

((σ(s, uε(s)

) − σ(s, u0(s)

))dW(s),V ε(s) − V 0(s)

)∣∣∣∣∣+

t∧τN∫0

∥∥σ(s, uε(s)

) − σ(s, u0(s)

)∥∥2LQ

ds

=: I1(t) + I2(t) + I3(t) + I4(t). (3.30)

3372 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

Taking the supremum up to time t ∧ τN in (3.30), and then taking the expectation, one obtains that

E

(sup

0≤s≤t∧τN

[∣∣V ε(s) − V 0(s)∣∣2 + 2

s∫0

∥∥V ε(l) − V 0(l)∥∥2

dl

])

≤ E

(I1(t) + I2(t) + sup

0≤s≤t

I3(s) + I4(t)). (3.31)

By (2.11), we have

EI1(t) ≤ 2E

( t∧τN∫0

(1

2

∥∥V ε(s) − V 0(s)∥∥2 + c

∥∥u0(s)∥∥4

L4 · ∣∣V ε(s) − V 0(s)∣∣2

)ds

)

≤ E

( t∧τN∫0

∥∥V ε(s) − V 0(s)∥∥2

ds

)

+ 2cE

( t∧τN∫0

sup0≤l≤s

∣∣V ε(l) − V 0(l)∣∣2 · ∥∥u0(s)

∥∥4L4ds

). (3.32)

By (2.9) and (2.10), we have

EI2(t) = 2√

εE

( t∧τN∫0

∣∣b(V ε(s),V ε(s),V ε(s) − V 0(s)

)∣∣ds

)

= 2√

εE

( t∧τN∫0

∣∣b(V ε(s),V ε(s),V 0(s)

)∣∣ds

)

≤ 4√

εE

( t∧τN∫0

∣∣V ε(s)∣∣ · ∥∥V ε(s)

∥∥ · ∥∥V 0(s)∥∥ds

)

≤ 2√

εE

( t∧τN∫0

∣∣V ε(s)∣∣2 · ∥∥V ε(s)

∥∥2ds +

t∧τN∫0

∥∥V 0(s)∥∥2

ds

). (3.33)

By the Burkholder–Davis–Gundy inequality and (2.16), we have

E

(sup

0≤s≤t

I3(s))

≤ 4E

( t∧τN∫ ∥∥σ(s, uε(s)

) − σ(s, u0(s)

)∥∥2LQ

· ∣∣V ε(s) − V 0(s)∣∣2

ds

) 12

0

R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390 3373

≤ 4KE

(sup

0≤s≤t∧τN

∣∣V ε(s) − V 0(s)∣∣ ·

( t∧τN∫0

∥∥uε(s) − u0(s)∥∥2

ds

) 12)

≤ 1

2E

(sup

0≤s≤t∧τN

∣∣V ε(s) − V 0(s)∣∣2

)

+ 8K2E

( t∧τN∫0

∥∥uε(s) − u0(s)∥∥2

ds

). (3.34)

By (2.16), we have

EI4(t) = E

( t∧τN∫0

∥∥σ(s, uε(s)

) − σ(s, u0(s)

)∥∥2LQ

ds

)

≤ KE

( t∧τN∫0

∥∥uε(s) − u0(s)∥∥2

ds

). (3.35)

Putting (3.31)–(3.35) together, and using Proposition 3.1 and Lemma 3.2 below, we have that

E

(sup

0≤s≤t∧τN

∣∣V ε(s) − V 0(s)∣∣2 +

t∧τN∫0

∥∥V ε(s) − V 0(s)∥∥2

ds

)

≤ cE

( t∧τN∫0

sup0≤l≤s

∣∣V ε(l) − V 0(l)∣∣2 · ∥∥u0(s)

∥∥4L4ds

)

+ c√

εE

( t∧τN∫0

∣∣V ε(s)∣∣2 · ∥∥V ε(s)

∥∥2ds +

t∧τN∫0

∥∥V 0(s)∥∥2

ds

)

+ cKE

( t∧τN∫0

∥∥uε(s) − u0(s)∥∥2

ds

)

≤ cE

( t∧τN∫0

sup0≤l≤s

∣∣V ε(l) − V 0(l)∣∣2 · ∥∥u0(s)

∥∥4L4ds

)+ cK(

√ε + ε).

Applying Gronwall’s inequality to g(t) = E(sup0≤s≤t∧τN|V ε(s) −V 0(s)|2) in the above inequal-

ity, we have

E

(sup

0≤s≤T ∧τN

∣∣V ε(s) − V 0(s)∣∣2 +

T ∧τN∫ ∥∥V ε(s) − V 0(s)∥∥2

ds

)

0

3374 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

≤ cK(√

ε + ε) exp

(c

T∫0

∥∥u0(s)∥∥4

L4ds

), (3.36)

which completes the proof. �It remains to establish the following estimate.

Lemma 3.2. Under the conditions (A.1)–(A.3), there exists a constant ε0 > 0 such that

sup0≤ε≤ε0

E

( T∫0

∣∣V ε(s)∣∣2 · ∥∥V ε(s)

∥∥2ds

)< ∞. (3.37)

Proof. Recall that

dV ε(t) + (AV ε(t) + B

(V ε(t), uε(t)

) + B(u0(t),V ε(t)

))dt = σ

(t, uε(t)

)dW(t).

By Itô’s formula, we have

d∣∣V ε(t)

∣∣4 = 2∣∣V ε(t)

∣∣2d∣∣V ε(t)

∣∣2 + d⟨∣∣V ε(·)∣∣2⟩

t

≤ 2∣∣V ε(t)

∣∣2(−2∥∥V ε(t)

∥∥2dt − 2

(B

(V ε(t), uε(t)

),V ε(t)

)dt

+ 2(σ(t, uε(t)

)dW(t),V ε(t)

) + ∥∥σ(t, uε(t)

)∥∥2LQ

dt)

+ 4∥∥σ

(t, uε(t)

)∥∥2LQ

· ∣∣V ε(t)∣∣2

dt.

Define τN = inf{t : |V ε(t)|4 + ∫ t

0 |V ε(s)|2 · ‖V ε(s)‖2ds > N}, then

∣∣V ε(t ∧ τN)∣∣4 + 4

t∧τN∫0

∣∣V ε(s)∣∣2 · ∥∥V ε(s)

∥∥2ds

≤ 4

∣∣∣∣∣t∧τN∫0

∣∣V ε(s)∣∣2(

B(V ε(s), uε(s)

),V ε(s)

)ds

∣∣∣∣∣ + 6

t∧τN∫0

∥∥σ(s, uε(s)

)∥∥2LQ

· ∣∣V ε(s)∣∣2

ds

+ 4

∣∣∣∣∣t∧τN∫0

∣∣V ε(s)∣∣2(

σ(s, uε(s)

)dW(s),V ε(s)

)∣∣∣∣∣. (3.38)

Taking the supremum up to time t ∧ τN in (3.38), and then taking the expectation, one obtains that

R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390 3375

E

(sup

0≤s≤t∧τN

[∣∣V ε(s)∣∣4 + 4

s∧τN∫0

∣∣V ε(l)∣∣2 · ∥∥V ε(l)

∥∥2ds

])

≤ 4E

( t∧τN∫0

∣∣V ε(s)∣∣2∣∣(B(

V ε(s), uε(s)),V ε(s)

)∣∣ds

)

+ 6E

( t∧τN∫0

∥∥σ(s, uε(s)

)∥∥2LQ

· ∣∣V ε(s)∣∣2

ds

)

+ 4E

(sup

0≤s≤t∧τN

∣∣∣∣∣s∫

0

∣∣V ε(l)∣∣2(

σ(l, uε(l)

)dW(l),V ε(l)

)∣∣∣∣∣)

=: I1(t) + I2(t) + I3(t). (3.39)

By the virtue of the properties of b(·, ·, ·), we have

I1(t) ≤ 4E

( t∧τN∫0

∣∣V ε(s)∣∣2∣∣(B(

V ε(s),V ε(s)), u0(s) + √

εV ε(s))∣∣ds

)

= 4E

( t∧τN∫0

∣∣V ε(s)∣∣2∣∣(B(

V ε(s),V ε(s)), u0(s)

)∣∣ds

)

≤ 4E

( t∧τN∫0

∣∣V ε(s)∣∣2 ·

(1

2

∥∥V ε(s)∥∥2 + c

∥∥u0(s)∥∥4

L4 · ∣∣V ε(s)∣∣2

)ds

)

≤ 2E

( t∧τN∫0

∣∣V ε(s)∣∣2 · ∥∥V ε(s)

∥∥2ds

)+ 4cE

( t∫0

sup0≤l≤s∧τN

∣∣V ε(l)∣∣4 · ∥∥u0(s)

∥∥4L4ds

). (3.40)

Recall that V ε(t) = (uε(t) − u0(t))/√

ε. By Lemma 3.1 and Proposition 3.1, we have

I2(t) ≤ 6KE

( t∧τN∫0

(1 + ∥∥uε(s)

∥∥2) · ∣∣V ε(s)∣∣2

ds

)

≤ 6KE

( t∧τN∫0

∣∣V ε(s)∣∣2

ds

)+ 12KE

( t∧τN∫0

(∥∥u0(s)∥∥2 + ε

∥∥V ε(s)∥∥2) · ∣∣V ε(s)

∣∣2ds

)

≤ cf,T ,K + 12KεE

( t∧τN∫ ∣∣V ε(s)∣∣2 · ∥∥V ε(s)

∥∥2ds

). (3.41)

0

3376 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

By the Burkholder–Davis–Gundy inequality and (2.16), we have

I3(t) ≤ cE

( t∧τN∫0

∣∣V ε(s)∣∣6 · ∥∥σ

(s, uε(s)

)∥∥2LQ

ds

) 12

≤ cE

(sup

0≤s≤t∧τN

∣∣V ε(s)∣∣2 ·

( t∧τN∫0

∣∣V ε(s)∣∣2 · ∥∥σ

(s, uε(s)

)∥∥2LQ

ds

) 12)

≤ 1

2E

(sup

0≤s≤t∧τN

∣∣V ε(s)∣∣4

)+ cKE

( t∧τN∫0

∣∣V ε(s)∣∣2 · (1 + ∥∥uε(s)

∥∥2)ds

)

≤ 1

2E

(sup

0≤s≤t∧τN

∣∣V ε(s)∣∣4

)+ 2cKE

( t∧τN∫0

∣∣V ε(s)∣∣2 · (1 + ∥∥u0(s)

∥∥2 + ε∥∥V ε(s)

∥∥2)ds

)

≤ 1

2E

(sup

0≤s≤t∧τN

∣∣V ε(s)∣∣4

)+ cT ,K + εcT ,KE

( t∧τN∫0

∣∣V ε(s)∣∣2 · ∥∥V ε(s)

∥∥2ds

). (3.42)

Substituting (3.40)–(3.42) into (3.39), we obtain

E

(sup

0≤s≤t∧τN

∣∣V ε(s)∣∣4 + (4 − cε)

t∧τN∫0

∣∣V ε(s)∣∣2 · ∥∥V ε(s)

∥∥2ds

)

≤ cf,T ,K + cE

( t∫0

∥∥u0(s)∥∥4

L4 · sup0≤l≤s∧τN

∣∣V ε(l)∣∣4

ds

). (3.43)

When ε < 2/c =: ε0, by Gronwall’s inequality, we have

E

(sup

0≤s≤T ∧τN

∣∣V ε(s)∣∣4 + 2

T ∧τN∫0

∣∣V ε(s)∣∣2 · ∥∥V ε(s)

∥∥2ds

)

≤ cf,T ,K exp

(c

T∫0

∥∥u0(s)∥∥4

L4ds

)< +∞. (3.44)

Letting N → ∞, we get

sup0≤ε≤ε0

E

(sup

0≤s≤T

∣∣V ε(s)∣∣4 + 2

T∫0

∣∣V ε(s)∣∣2 · ∥∥V ε(s)

∥∥2ds

)< +∞, (3.45)

which is stronger than the desired result. The proof is complete. �

R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390 3377

4. Moderate deviations

In this part, we will prove that 1√ελ(ε)

(uε − u0) satisfies an LDP on C([0, T ]; H) ∩L2([0, T ]; V ) with λ(ε) satisfying (1.5). This special type of LDP is usually called the mod-erate deviation principle of uε (cf. [8]).

4.1. Large deviation principle

In this section, we will recall the general criteria for a large deviation principle given in [3].Let (Ω, F, P) be a probability space with an increasing family {Ft}0≤t≤T of the sub-σ -fields

of F satisfying the usual conditions. Let E be a Polish space with the Borel σ -field B(E).

Definition 4.1 (Rate function). A function I : E → [0, ∞] is called a rate function on E , if for each M < ∞, the level set {x ∈ E : I (x) ≤ M} is a compact subset of E .

Definition 4.2 (Large deviation principle). Let I be a rate function on E . A family {Xε} of E -valued random elements is said to satisfy the large deviation principle on E with rate func-tion I , if the following two conditions hold.

(a) (Large deviation upper bound) For each closed subset F of E ,

lim supε→0

ε logP(Xε ∈ F

) ≤ − infx∈F

I (x).

(b) (Large deviation lower bound) For each open subset G of E ,

lim infε→0

ε logP(Xε ∈ G

) ≥ − infx∈G

I (x).

The Cameron–Martin space associated with the Wiener process {W(t), t ∈ [0, T ]} is given by

H0 :={

h : [0, T ] → H0; h is absolutely continuous and

T∫0

∣∣h(s)∣∣20ds < +∞

}. (4.46)

The space H0 is a Hilbert space with inner product

〈h1, h2〉H0 :=T∫

0

⟨h1(s), h2(s)

⟩0ds.

Let A denote the class of H0-valued {Ft }-predictable processes φ belonging to H0 a.s. Let SN = {h ∈ H0;

∫ T

0 |h(s)|20ds ≤ N}. The set SN endowed with the weak topology is a Polish space. Define AN = {φ ∈A; φ(ω) ∈ SN, P-a.s.}.

3378 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

Recall the following result from Budhiraja and Dupuis [3].

Theorem 4.3. (See [3].) For ε > 0, let Γ ε be a measurable mapping from C([0, T ]; H) into E . Let Xε := Γ ε(W(·)). Suppose that {Γ ε}ε>0 satisfies the following assumptions: there exists a measurable map Γ 0 : C([0, T ]; H) → E such that

(a) for every N < +∞ and any family {hε; ε > 0} ⊂ AN satisfying that hε converge in distri-bution as SN -valued random elements to h as ε → 0, Γ ε(W(·) + 1√

ε

∫ ·0 hε(s)ds) converges

in distribution to Γ 0(∫ ·

0 h(s)ds) as ε → 0;(b) for every N < +∞, the set

{Γ 0

( ·∫0

h(s)ds

); h ∈ SN

}

is a compact subset of E .

Then the family {Xε}ε>0 satisfies a large deviation principle in E with the rate function I given by

I (g) := inf{h∈H0; g=Γ 0(

∫ ·0 h(s)ds)}

{1

2

T∫0

∣∣h(s)∣∣20ds

}, g ∈ E, (4.47)

with the convention inf{∅} = ∞.

4.2. Skeleton equations

Let H be a separable Hilbert space. Given p > 1, α ∈ (0, 1), let Wα,p([0, T ]; H) be the Sobolev space of all u ∈ Lp([0, T ]; H) such that

T∫0

T∫0

‖u(t) − u(s)‖p

H

|t − s|1+αpdtds < ∞

endowed with the norm

‖u‖p

Wα,p([0,T ];H):=

T∫0

∥∥u(t)∥∥p

Hdt +

T∫0

T∫0

‖u(t) − u(s)‖p

H

|t − s|1+αpdtds.

The following result represents a variant of the criteria for compactness proved in [18](Sect. 5, Ch. I) and [22] (Sect. 13.3).

R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390 3379

Lemma 4.1. Let H0 ⊂H ⊂ H1 be Banach spaces, H0 and H1 reflexive, with compact embedding of H0 in H. For p ∈ (1, ∞) and α ∈ (0, 1), let Λ be the space

Λ = Lp([0, T ];H0

) ∩ Wα,p([0, T ];H1

)endowed with the natural norm. Then the embedding of Λ in Lp([0, T ]; H) is compact.

We begin by introducing the map Γ 0 that will be used to define the rate function and also used for verification of conditions in Theorem 4.3.

For any h ∈H0, consider the deterministic integral equation

dXh(t) + (AXh(t) + B

(Xh(t), u0(t)

) + B(u0(t),Xh(t)

))dt = σ

(t, u0(t)

)h(t)dt, (4.48)

with initial value Xh(0) = 0.

Proposition 4.4. Under the conditions (A.1)–(A.3), for any h ∈ H0, Eq. (4.48) admits a unique solution Xh in C([0, T ]; H) ∩ L2([0, T ]; V ). Moreover, for any N > 0 and α ∈ (0, 1/2), there exist constants cN,T and cα,N such that

suph∈SN

{sup

0≤t≤T

∣∣Xh(t)∣∣2 +

T∫0

∥∥Xh(s)∥∥2

ds

}≤ cN,T , (4.49)

and

suph∈SN

∥∥Xh∥∥

Wα,2([0,T ];V ′) ≤ cα,N,T . (4.50)

Proof. Using the classical Galerkin approximation scheme, the existence and uniqueness of the solution can be proved similarly as in the case of the Navier–Stokes equation, but much simpler.

Here, we will prove (4.49) and (4.50).

Proof of (4.49): For any h ∈ SN , by (2.9), (2.11) and (2.16), we have

∣∣Xh(t)∣∣2 + 2

t∫0

∥∥Xh(s)∥∥2

ds

= −2

t∫0

(B

(Xh(s), u0(s)

),Xh(s)

)ds + 2

t∫0

(σ(Xh(s)

)h(s),Xh(s)

)ds

≤t∫

0

∥∥Xh(s)∥∥2

ds + c

t∫0

∣∣Xh(s)∣∣2∥∥u0(s)

∥∥44ds + 2

t∫0

∣∣h(s)∣∣0

∥∥σ(Xh(s)

)∥∥LQ

∣∣Xh(s)∣∣ds

≤t∫ ∥∥Xh(s)

∥∥2ds + c

t∫ ∣∣Xh(s)∣∣2∥∥u0(s)

∥∥44ds

0 0

3380 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

+ 1

2K

t∫0

∥∥σ(Xh(s)

)∥∥2LQ

ds + 2K

t∫0

∣∣h(s)∣∣20 · ∣∣Xh(s)

∣∣2ds

≤ 3

2

t∫0

∥∥Xh(s)∥∥2

ds +t∫

0

∣∣Xh(s)∣∣2(

c∥∥u0(s)

∥∥4L4 + 2K

∣∣h(s)∣∣20

)ds + T

2,

where K is the constant appeared in (2.16). Then

sup0≤s≤t

∣∣Xh(s)∣∣2 + 1

2

t∫0

∥∥Xh(s)∥∥2

ds ≤t∫

0

sup0≤l≤s

∣∣Xh(l)∣∣2(

c∥∥u0(s)

∥∥4L4 + 2K

∣∣h(s)∣∣20

)ds + T

2.

Since h ∈ SN , by Gronwall’s inequality and Lemma 3.1(iii), we get

sup0≤t≤T

∣∣Xh(t)∣∣2 + 1

2

T∫0

∥∥Xh(s)∥∥2

ds ≤ T

2exp

{ T∫0

(c∥∥u0(s)

∥∥4L4 + 2K

∣∣h(s)∣∣20

)ds

}< ∞,

which yields (4.49).

Proof of (4.50): Next we will prove (4.50). Notice that

Xh(t) = −t∫

0

AXh(s)ds −t∫

0

B(Xh(s), u0(s)

)ds

−t∫

0

B(u0(s),Xh(s)

)ds +

t∫0

σ(s, u0(s)

)h(s)ds

=: I1(t) + I2(t) + I3(t) + I4(t). (4.51)

Using the same arguments as in the proof of Theorem 3.1 in [11], we have

‖I1‖2Wα,2([0,T ;V ′]) ≤ L1. (4.52)

By (2.10) and the Cauchy–Schwarz inequality, we have, for any 0 ≤ s ≤ t ≤ T ,

∥∥∥∥∥t∫

s

B(Xh(l), u0(l)

)dl

∥∥∥∥∥2

V ′≤

( t∫s

∥∥B(Xh(l), u0(l)

)∥∥V ′dl

)2

≤( t∫

s

c∥∥Xh(l)

∥∥∥∥u0(l)∥∥dl

)2

≤ c

( T∫ ∥∥Xh(l)∥∥2

dl

)( t∫ ∥∥u0(l)∥∥2

dl

).

0 s

R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390 3381

Thus,

T∫0

∥∥I2(s)∥∥2

V ′ds ≤ cT

( T∫0

∥∥Xh(l)∥∥2

dl

)( T∫0

∥∥u0(l)∥∥2

dl

)< +∞, (4.53)

and

T∫0

T∫0

‖I2(t) − I2(s)‖2V ′

|t − s|1+2αdsdt ≤ cT

( T∫0

∥∥Xh(l)∥∥2

dl

)×

T∫0

T∫0

t∫s

‖u0(l)‖2

|t − s|1+2αdldsdt. (4.54)

By elementary application of Cauchy–Schwarz inequality and Fubini theorem, there exists cα,T > 0 such that

T∫0

T∫0

t∫s

‖u0(l)‖2

|t − s|1+2αdldsdt ≤ cα,T

T∫0

∥∥u0(l)∥∥2

dl. (4.55)

Combining (4.53), (4.54) and (4.55), we have

‖I2‖2Wα,2([0,T ;V ′]) ≤ L2. (4.56)

Similarly, we also have

‖I3‖2Wα,2([0,T ;V ′]) ≤ L3. (4.57)

It remains to deal with the last term I4. Since h ∈ SN , by (2.16), we have

T∫0

∥∥∥∥∥t∫

0

σ(s, u0(s)

) · h(s)ds

∥∥∥∥∥2

V ′dt ≤

T∫0

∣∣∣∣∣t∫

0

∥∥σ(s, u0(s)

)∥∥LQ

∣∣h(s)∣∣0ds

∣∣∣∣∣2

dt

≤ T

T∫0

∥∥σ(s, u0(s)

)∥∥2LQ

ds

T∫0

∣∣h(s)∣∣20ds

≤ T NK

T∫0

(1 + ∥∥u0(s)

∥∥2)ds,

and∥∥∥∥∥t∫σ(l, u0(l)

) · h(l)dl

∥∥∥∥∥2

′≤

t∫ ∥∥σ(l, u0(l)

)∥∥2LQ

dl ·t∫ ∣∣h(l)

∣∣20dl ≤ NK

t∫ (1 + ∥∥u0(l)

∥∥2)dl.

s V s s s

3382 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

Similar to (4.56), the above two inequalities imply

‖I4‖2Wα,2([0,T ;V ′]) ≤ L4. (4.58)

By (4.52), (4.56), (4.57) and (4.58), we obtain (4.50). The proof is complete. �For h ∈H0, set

Γ 0

( ·∫0

h(s)ds

):= Xh, (4.59)

where Xh is the solution of Eq. (4.48).

Proposition 4.5. Under the conditions (A.1)–(A.3), for every positive number N < ∞, the family

KN :={

Γ 0

( ·∫0

h(s)ds

); h ∈ SN

}

is compact in C([0, T ]; H) ∩ L2([0, T ]; V ).

Proof. Let {Xhn = Γ 0(∫ ·

0 hn(s)ds); n ≥ 1} be a sequence of elements in KN . The estimates (4.49) and (4.50) enable us to assert that there exist a subsequence {n′} and h ∈ SN such that

(a) hn′ → h in SN as n′ → ∞,(b) Xhn′ → Xh in L2([0, T ]; V ) weakly,(c) Xhn′ → Xh in L∞([0, T ]; H) weak-star,(d) Xhn′ → Xh in L2([0, T ]; H) strongly.

Using the same argument as in the proof of [21, Theorem 3.1, p. 191], we can conclude that Xh = Γ 0(

∫ ·0 h(s)ds).

Next, we will prove that Xhn′ → Xh in C([0, T ]; H) ∩ L2([0, T ]; V ). Using (2.9) and (2.11), we obtain

∣∣Xhn′ (t) − Xh(t)∣∣2 + 2

t∫0

∥∥Xhn′ (s) − Xh(s)∥∥2

ds

= −2

t∫0

(B

(Xhn′ (s) − Xh(s), u0(s)

),Xhn′ (s) − Xh(s)

)ds

− 2

t∫ (B

(u0(s),Xhn′ (s) − Xh(s)

),Xhn′ (s) − Xh(s)

)ds

0

R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390 3383

+ 2

t∫0

(σ(s, u0(s)

)(hn′(s) − h(s)

),Xhn′ (s) − Xh(s)

)ds

≤t∫

0

∥∥Xhn′ (s) − Xh(s)∥∥2

ds + c

t∫0

∥∥u0(s)∥∥4

L4 · ∣∣Xhn′ (s) − Xh(s)∣∣2

ds

+ 2

t∫0

∥∥σ(s, u0(s)

)∥∥LQ

· ∣∣hn′(s) − h(s)∣∣0 · ∣∣Xhn′ (s) − Xh(s)

∣∣ds.

By the Cauchy–Schwarz inequality and (2.16), we have

sup0≤s≤T

∣∣Xhn′ (s) − Xh(s)∣∣2 +

T∫0

∥∥Xhn′ (s) − Xh(s)∥∥2

ds

≤ c

T∫0

∥∥u0(s)∥∥4

L4 · ∣∣Xhn′ (s) − Xh(s)∣∣2

ds

+ 2

(K

T∫0

(1 + ∥∥u0(s)

∥∥2) · ∣∣Xhn′ (s) − Xh(s)∣∣2

ds

) 12

·( T∫

0

∣∣hn′(s) − h(s)∣∣20ds

) 12

≤ c

T∫0

∥∥u0(s)∥∥4

L4 · ∣∣Xhn′ (s) − Xh(s)∣∣2

ds

+ 4√

KN

( T∫0

(1 + ∥∥u0(s)

∥∥2) · ∣∣Xhn′ (s) − Xh(s)∣∣2

ds

) 12

,

where we have used the fact

T∫0

∣∣hn′(s) − h(s)∣∣20ds ≤ 2

T∫0

∣∣hn′(s)∣∣20ds + 2

T∫0

∣∣h(s)∣∣20ds ≤ 4N,

since hn′ and h are in SN .Notice that Xhn′ → Xh in L2([0, T ]; H) strongly, for any ε > 0,

Leb(s ∈ [0, T ]; ∣∣Xhn′ (s) − Xh(s)

∣∣ > ε) → 0, as n′ → ∞,

where Leb(A) means the Lebesgue measure of A ∈ B(R). With the help of estimates in Lemma 3.1(iii), (4.49) and the dominant control theorem, we have

3384 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

sup0≤s≤T

∣∣Xhn′ (s) − Xh(s)∣∣2 +

T∫0

∥∥Xhn′ (s) − Xh(s)∥∥2

ds → 0, as n′ → ∞.

This means that Xhn′ → Xh in C([0, T ]; H) ∩ L2([0, T ]; V ), and KN is compact inC([0, T ]; H) ∩ L2([0, T ]; V ). The proof is complete. �4.3. Moderate deviation principle

We are now ready to state the second main result.Let I be the rate function (4.47) with Γ 0 given in (4.59).

Theorem 4.6 (Moderate deviation principle). Under the conditions (A.1)–(A.3), (uε − u0)/

(√

ελ(ε)) obeys an LDP on C([0, T ]; H) ∩ L2([0, T ]; V ) with speed λ2(ε) and with rate func-tion I , more precisely, it holds that

(a) for each closed subset F of C([0, T ]; H) ∩ L2([0, T ]; V ),

lim supε→0

1

λ2(ε)logP

(uε − u0

√ελ(ε)

∈ F

)≤ − inf

x∈FI (x);

(b) for each open subset G of C([0, T ]; H) ∩ L2([0, T ]; V ),

lim infε→0

1

λ2(ε)logP

(uε − u0

√ελ(ε)

∈ G

)≥ − inf

x∈GI (x).

Proof. Let Zε = (uε − u0)/(√

ελ(ε)). Then Zε satisfies the following SPDE:

dZε(t) + (AZε(t) + B

(Zε(t), u0(t) + √

ελ(ε)Zε(t)) + B

(u0(t),Zε(t)

))dt

= λ−1(ε)σ(t, u0(t) + √

ελ(ε)Zε(t))dW(t), (4.60)

with initial value Zε(0) = 0. This equation admits a unique solution Zε = Γ ε(W(·)), where Γ ε

stands for the solution functional from C([0, T ]; H) into C([0, T ]; H) ∩ L2([0, T ]; V ).According to Theorem 4.3, we need to prove that two conditions in Theorem 4.3 are fulfilled.

Condition (b) has been established in Proposition 4.5. The verification of condition (a) will be given by Proposition 4.7 below. �

For any φε ∈A, consider

dXε(t) + (AXε(t) + B

(Xε(t), u0(t) + √

ελ(ε)Xε(t)) + B

(u0(t),Xε(t)

))dt

= λ−1(ε)σ(t, u0(t) + √

ελ(ε)Xε(t))dW(t) + σ

(t, u0(t) + √

ελ(ε)Xε(t))φε(t)dt, (4.61)

with initial value Xε(0) = 0. Then it follows from the definition Γ ε that

Xε = Γ ε

(W(·) + λ(ε)

·∫φε(s)ds

).

0

R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390 3385

Lemma 4.2. Under the conditions (A.1)–(A.3), for any family {φε, ε > 0} ⊂ AN , there exists ε0 > 0 such that

sup0≤ε≤ε0

E

(sup

0≤t≤T

∣∣Xε(t)∣∣2 +

T∫0

∥∥Xε(t)∥∥2

dt

)≤ cN,T , (4.62)

and

sup0≤ε≤ε0

E(∥∥Xε

∥∥2Wα,2([0,T ];V ′)

) ≤ cα,N,T . (4.63)

Thus the family {Xε, 0 ≤ ε ≤ ε0} is tight in L2([0, T ]; H), by Lemma 4.1.

Proof. Define τM := inf{t : |Xε(t)|2 + ∫ t

0 ‖Xε(s)‖2ds > M}. By Itô’s formula and (2.9), we have

∣∣Xε(t ∧ τM)∣∣2 + 2

t∧τM∫0

∥∥Xε(s)∥∥2

ds

= −2

t∧τM∫0

(B

(Xε(s), u0(s)

),Xε(s)

)ds

+ 2λ−1(ε)

t∧τM∫0

(σ(s, u0(s) + √

ελ(ε)Xε(s))dW(s),Xε(s)

)

+ 2

t∧τM∫0

(σ(s, u0(s) + √

ελ(ε)Xε(s))φε(s),Xε(s)

)ds

+ λ−2(ε)

t∧τM∫0

∥∥σ(s, u0(s) + √

ελ(ε)Xε(s))∥∥2

LQds.

Taking the supremum up to time t ∧ τM in above equation, and then taking the expectation, one obtains that

E

(sup

0≤s≤t∧τM

∣∣Xε(s)∣∣2 + 2

t∧τM∫0

∥∥Xε(s)∥∥2

ds

)

≤ 2E

( t∧τM∫ ∣∣(B(Xε(s), u0(s)

),Xε(s)

)∣∣ds

)

0

3386 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

+ 2λ−1(ε)E

(sup

0≤s≤t∧τM

s∫0

(σ(s, u0(l) + √

ελ(ε)Xε(l))dW(l),Xε(l)

))

+ 2E

( t∧τM∫0

∣∣(σ (s, u0(s) + √

ελ(ε)Xε(s))φε(s),Xε(s)

)∣∣ds

)

+ λ−2(ε)E

( t∧τM∫0

∥∥σ(s, u0(s) + √

ελ(ε)Xε(s))∥∥2

LQds

)

=: I1(t) + I2(t) + I3(t) + I4(t). (4.64)

By (2.11), we have

∣∣I1(t)∣∣ ≤ E

( t∧τM∫0

∥∥Xε(s)∥∥2

ds

)+ c

t∫0

E

(sup

0≤l≤s∧τM

∣∣Xε(l)∣∣2

)· ∥∥u0(s)

∥∥4L4ds. (4.65)

By the Burkholder–Davis–Gundy inequality, we have

∣∣I2(t)∣∣ ≤ cλ−1(ε)E

( t∧τM∫0

∥∥σ(s, u0(s) + √

ελ(ε)Xε(s))∥∥2

LQ· ∣∣Xε(s)

∣∣2ds

) 12

≤ cλ−1(ε)E

(sup

0≤s≤t∧τM

∣∣Xε(s)∣∣ ·

( t∧τM∫0

2K(1 + ∥∥u0(s)

∥∥2 + ελ2(ε)∥∥Xε(s)

∥∥2)ds

) 12)

≤ c

2λ−1(ε)E

(sup

0≤s≤t∧τM

∣∣Xε(s)∣∣2

)

+ cKλ−1(ε)E

( t∧τM∫0

(1 + ∥∥u0(s)

∥∥2 + ελ2(ε)∥∥Xε(s)

∥∥2)ds

)

≤ c

2λ−1(ε)E

(sup

0≤s≤t∧τM

∣∣Xε(s)∣∣2

)+ cKελ(ε)E

( t∧τM∫0

∥∥Xε(s)∥∥2

ds

)+ cK,T . (4.66)

For the third term I3, we have

∣∣I3(t)∣∣ ≤ 2E

( t∧τM∫0

∥∥σ(s, u0(s) + √

ελ(ε)Xε(s))∥∥

LQ· ∣∣φε(s)

∣∣0 · ∣∣Xε(s)

∣∣ds

)

≤ 2E

(sup

0≤s≤t∧τM

∣∣Xε(s)∣∣ ·

t∧τM∫ ∥∥σ(s, u0(s) + √

ελ(ε)Xε(s))∥∥

LQ· ∣∣φε(s)

∣∣0ds

)

0

R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390 3387

≤ 1

2E

(sup

0≤s≤t∧τM

∣∣Xε(s)∣∣2

)

+ 2E

( t∧τM∫0

∥∥σ(s, u0(s) + √

ελ(ε)Xε(s))∥∥ · ∣∣φε(s)

∣∣0ds

)2

≤ 1

2E

(sup

0≤s≤t∧τM

∣∣Xε(s)∣∣2

)

+ 2E

( t∧τM∫0

∥∥σ(s, u0(s) + √

ελ(ε)Xε(s))∥∥2

ds ·t∧τM∫0

∣∣φε(s)∣∣20ds

)

≤ 1

2E

(sup

0≤s≤t∧τM

∣∣Xε(s)∣∣2

)+ 2KNελ2(ε)E

( t∧τM∫0

∥∥Xε(s)∥∥2

ds

)+ cf,T , (4.67)

where we have used the fact of φε ∈ AN , (2.16) and Lemma 3.1 in the last inequality.For the last term I4,

∣∣I4(t)∣∣ ≤ 2λ−2(ε)KE

( t∫0

(1 + ∥∥u0(s)

∥∥2 + ελ2(ε)∥∥Xε(s)

∥∥2)ds

). (4.68)

Putting (4.64)–(4.68) together, we have

E

((sup

0≤s≤t∧τM

(1 − c1λ

−1(ε))∣∣Xε(s)

∣∣2)

+ (1 − cK,T

(ελ2(ε) + ελ(ε) + ε

)) t∧τM∫0

∥∥Xε(s)∥∥2

ds

)

≤ cK,T + c2

t∫0

∥∥u0(s)∥∥4

L4 ·E(

sup0≤l≤s∧τM

∣∣Xε(l)∣∣2

)ds.

By Gronwall’s inequality and Lemma 3.1(iii), in view of (1.5), there exists ε0 > 0 such that

sup0≤ε≤ε0

E

(sup

0≤s≤t∧τM

∣∣Xε(s)∣∣2 +

t∧τM∫0

∥∥Xε(s)∥∥2

ds

)≤ c1 exp

(c2

t∫0

∥∥u0(s)∥∥4

L4ds

)< cN,T .

Letting M → ∞, we get (4.62).(2) The proof of (4.63) is very similar to that of the deterministic case (4.50), we omit it here.The proof is complete. �

Proposition 4.7. Assume that (A.1)–(A.3) hold. For every fixed N ∈ N, let φε , φ ∈ AN be such that φε converges in distribution to φ as ε → 0. Then

3388 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

Γ ε

(W(·) + λ(ε)

·∫0

φε(s)ds

)converges in distribution to Γ 0

( ·∫0

φ(s)ds

)

in C([0, T ]; H) ∩ L2([0, T ]; V ) as ε → 0.

Proof. Note that Xε = Γ ε(W(·) + λ(ε) ∫ ·

0 φε(s)ds). Using similar arguments as in the proof of Theorem 3.1 in [11], we can prove that {Xε, 0 < ε ≤ ε0} is tight in C([0, T ]; D(A−�)), here �appears in (2.14). Combining this with Lemma 4.2, we know that {Xε, ε ∈ (0, ε0)} is tight in L2([0, T ]; H) ∩ C([0, T ]; D(A−�)).

By Lemma 4.2, it is easy to see that there exists a unique solution Y ε ∈ C([0, T ]; H) ∩L2([0, T ]; V ) satisfying

dY ε(t) + AYε(t)dt = λ−1(ε)σ(t, u0(t) + √

ελ(ε)Xε(t))dW(t),

with initial value Y ε(0) = 0, and

limε→0

[E sup

t∈[0,T ]∣∣Y ε(t)

∣∣2 +E

T∫0

∥∥Y ε(t)∥∥2

dt

]= 0. (4.69)

Set

Ξ := (L2([0, T ];H ) ∩ C

([0, T ];D(A−�

)), SN,C

([0, T ];H ) ∩ L2([0, T ];V )).

Then the family {(Xε, φε, Y ε), ε ∈ (0, ε0)} is tight in Ξ . Let (X, φ, 0) be any limit point of {(Xε, φε, Y ε), ε ∈ (0, ε0)}. We will show that X has the same law as Γ 0(

∫ ·0 φ(s)ds), and actu-

ally Xε convergence in distribution to X in the space C([0, T ]; H) ∩ L2([0, T ]; V ). This will complete the proof of Proposition 4.7.

In fact, by the Skorokhod representation theorem, there exist a stochastic basis(Ω, F, {Ft }t∈[0,T ], P) and, on this basis, Ξ -valued random variables (X, φ, 0), (Xε, φε, Y ε), ε ∈ (0, ε0), such that (Xε, φε, Y ε) (respectively (X, φ, 0)) has the same law as (Xε, φε, Y ε)

(respectively (X, φ, 0)), and (Xε, φε, Y ε) → (X, φ, 0) in Ξ , P-a.s.From the equation satisfied by (Xε, φε, Y ε), we see that (Xε, φε, Y ε) satisfies the following

integral equation

d(Xε(t) − Y ε(t)

) + A(Xε(t) − Y ε(t)

)dt

+ (B

(Xε(t), u0(t) + √

ελ(ε)Xε(t)) + B

(u0(t), Xε(t)

))dt

= σ(t, u0(t) + √

ελ(ε)Xε(t)) ˙φ ε

(t)dt,

and

P(Xε − Y ε ∈ C

([0, T ];H ) ∩ L2([0, T ];V ))= P

(Xε − Y ε ∈ C

([0, T ];H ) ∩ L2([0, T ];V ))= 1.

R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390 3389

Let Ω0 be the subset of Ω such that (Xε, φε, Y ε) → (X, φ, 0) in Ξ , then P(Ω0) = 1. For any fixed ω ∈ Ω0, actually the following stronger convergence holds

supt∈[0,T ]

∣∣Xε(ω, t) − X(ω, t)∣∣2 +

T∫0

∥∥Xε(ω, t) − X(ω, t)∥∥2

dt → 0 as ε → 0. (4.70)

Set Zε = Xε − Y ε , then Zε(ω) ∈ C([0, T ]; H) ∩ L2([0, T ]; V ), and Zε(ω) satisfies

dZε(t) + AZε(t)dt

+ (B

([Zε(t) + Y ε(t)

], u0(t) + √

ελ(ε)[Zε(t) + Y ε(t)

]) + B(u0(t),

[Zε(t) + Y ε(t)

]))dt

= σ(t, u0(t) + √

ελ(ε)[Zε(t) + Y ε(t)

]) ˙φ ε(t)dt,

with initial value Zε(0) = 0.Since

limε→0

[sup

t∈[0,T ]∣∣Y ε(ω, t)

∣∣2 +T∫

0

∥∥Y ε(ω, t)∥∥2

dt

]= 0,

using the above equation of Zε and by the similar arguments as in the proof of Proposition 4.5, we can show that

limε→0

[sup

t∈[0,T ]∣∣Xε(ω, t) − X(ω, t)

∣∣2 +T∫

0

∥∥Xε(ω, t) − X(ω, t)∥∥2

dt

]= 0, (4.71)

where

X(t) = −t∫

0

AX(s)ds −t∫

0

(B

(X(s), u0(s)

) + B(u0(s), X(s)

))ds +

t∫0

σ(s, u0(s)

) ˙φ(s)ds.

Hence X = X = Γ 0(∫ ·

0˙φ(s)ds), and X has the same law as Γ 0(

∫ ·0 φ(s)ds). Since Xε and Xε

have the same law on C([0, T ]; H) ∩ L2([0, T ]; V ), (4.71) further implies that Γ ε(W(·) +λ(ε)

∫ ·0 φε(s)ds) converges in distribution to Γ 0(

∫ ·0 φ(s)ds) as ε → 0.

The proof is complete. �Acknowledgments

The authors are grateful to the anonymous referees for their valuable comments and sugges-tions. This work was supported by the National Natural Science Foundation of China (NSFC) (No. 11431014, No. 11301498, No. 11401557), and the Fundamental Research Funds for the Central Universities (No. WK 0010000033).

3390 R. Wang et al. / J. Differential Equations 258 (2015) 3363–3390

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