moderate deviations for a stochastic heat equation with spatially...

Acta Appl Math (2015) 139:59–80DOI 10.1007/s10440-014-9969-x

Moderate Deviations for a Stochastic Heat Equationwith Spatially Correlated Noise

Yumeng Li · Ran Wang · Shuguang Zhang

Received: 19 March 2014 / Accepted: 12 September 2014 / Published online: 18 September 2014© Springer Science+Business Media Dordrecht 2014

Abstract In this paper, we proved a central limit theorem and established a moderate de-viation principle for a perturbed stochastic heat equations defined on [0, T ] × [0,1]d . Thisequation is driven by a Gaussian noise, white in time and correlated in space.

Keywords Stochastic heat equation · Freidlin-Wentzell’s large deviation · Moderatedeviations · Central limit theorem

Mathematics Subject Classification 60H15 · 60F05 · 60F10

1 Introduction

Since the work of Freidlin and Wentzell [17], the theory of small perturbation large devia-tions for stochastic (partial) differential equation has been extensively developed (see [10,12, 13]). The large deviation principle (LDP) for stochastic reaction-diffusion equationsdriven by the space-time white noise was first obtained by Freidlin [16] and later by Sow-ers [31], Chenal and Millet [6], Carrai and Röckner [4] and other authors. An LDP for astochastic heat equation driven by a Gaussian noise, white in time and correlated in spacewas proved by Márquez-Carreras and Sarrà [26].

Y. Li · S. ZhangDepartment of Statistics and Finance, University of Science and Technology of China, Hefei,Anhui Province 230026, China

Y. Lie-mail: [email protected]

S. Zhange-mail: [email protected]

R. Wang (B)School of Mathematical Sciences, University of Science and Technology of China, Hefei,Anhui Province 230026, Chinae-mail: [email protected]

http://crossmark.crossref.org/dialog/?doi=10.1007/s10440-014-9969-x&domain=pdf

mailto:[email protected]



60 Y. Li et al.

Like the large deviations, the moderate deviation problems arise in the theory of statisticalinference quite naturally. The moderate deviation principle (MDP) can provide us with therate of convergence and a useful method for constructing asymptotic confidence intervals,see [14, 20, 21, 23, 24] and references therein. Results on the MDP for processes withindependent increments were obtained in De Acosta [1], Chen [5] and Ledoux [25]. Thestudy of the MDP estimates for other processes has been carried out as well, e.g., Wu [34]for Markov processes, Guillin and Liptser [22] for diffusion processes, Budhiraja et al. [3]for stochastic differential equations with jumps, and references therein.

Wang and Zhang [33] proved a central limit theorem and established a moderate devia-tion principle for stochastic reaction-diffusion equations on driven by the space-time whitenoise [0, T ] × [0,1]. However the high dimensional case is much more complicated, onedifficulty comes from the more complicated stochastic integral in the sense of Dalang [8],the other one comes from the Hölder continuity of the Green functions in high dimension.

In this paper, we shall study it in this paper. More precisely, we shall prove a central limittheorem and a moderate deviation principle for a perturbed stochastic heat equations definedon [0, T ]× [0,1]d . This equation is driven by a Gaussian noise, white in time and correlatedin space.

Let us give the framework taken from Márquez-Carreras and Sarrà [26].Consider the following perturbed d-dimensional spatial stochastic heat equation on the

compact set [0,1]d ,

⎧⎪⎨

⎪⎩

Luε(t, x) = √εα(uε(t, x))F (t, x) + β(uε(t, x)), t ≥ 0, x ∈ [0,1]d ,

uε(t, x) = 0, x ∈ ∂([0,1]d),uε(0, x) = 0, x ∈ [0,1]d ,

(1)

with ε > 0, L = ∂∂t

− � where � is the Laplacian on Rd and ∂([0,1]d) is the boundary of

[0,1]d .Assume that the coefficients satisfy the following conditions:

(C) the functions α and β are Lipschitz,

i.e., there exists some constant K such that

|α(x) − α(y)| ≤ K|x − y|, |β(x) − β(y)| ≤ K|x − y| (2)

for all x, y ∈ R. The noise F = {F(ϕ),ϕ : Rd+1 → R} is an L2(Ω,F,P)-valued Gaussianprocess with mean zero and covariance functional given by

J (ϕ,ψ) =∫

R+ds

∫

Rd

dx

∫

Rd

dyϕ(s, x)f (x − y)ψ(s, y), (3)

and f : Rd → R+ is a continuous symmetric function on Rd − {0} such that there exists a

non-negative tempered measure λ on Rd , whose Fourier transform is f . The function in (3)

is said to be a covariance functional if all these assumptions are satisfied. Then, in addition,

J (ϕ,ψ) =∫

R+ds

∫

Rd

λ(dξ)Fϕ(s, ·)(ξ)Fψ(s, ·)(ξ),

where F is the Fourier transform and z is the conjugate complex of z. In (3), we could alsowork with a non-negative and non-negative definite tempered measure, therefore symmetric,

Moderate Deviations for a Stochastic Heat Equation 61

instead of the function f but, in this case, all the notation over the sets appearing in theintegrals is becoming tedious. In this paper, moreover, we assume the following hypothesison the measure λ:

(Hη)

∫

Rd

λ(dξ)

(1 + ‖ξ‖2)η< +∞, (4)

for some η ∈ (0,1]. For instance, the function f (x) = ‖x‖−κ , κ ∈ (0, d), satisfies (4). As inDalang [8], the Gaussian process F can be extended to a worthy martingale measure, in thesense of Walsh [32],

M = {Mt(A), t ∈R+, A ∈ Bb

(R

d)}

,

where Bb(Rd) denotes the collection of all bounded Borel measurable sets in R

d .Under the assumptions (C) and (Hη), following the approach of Walsh [32], one can

prove that there is a unique solution to Eq. (1) given by the following mild form

uε(t, x) = √ε

∫ t

0

∫

[0,1]dGt−s(x, y)α

(uε(s, y)

)F(dsdy)

+∫ t

0ds

∫

[0,1]ddyGt−s(x, y)β

(uε(s, y)

), (5)

where Gt(x, y) is the Green kernel associated with the heat equation on [0,1]d :

⎧⎪⎨

⎪⎩

∂∂t

Gt (x, y) = �xGt(x, y), t ≥ 0, x, y ∈ [0,1]d ,Gt (x, y) = 0, x ∈ ∂([0,1]d),G0(x, y) = δ(x − y).

(6)

The stochastic integral in (5) is defined with respect to the Ft -martingale measure M . DenoteDd

T := [0, T ] × [0,1]d . If d > 1, the evolution equation can not be driven by the Browniansheet because G does not belong to L2(Dd

T ) and we need to work with a smoother noise.The study of existence and uniqueness of solution to Eq. (5) on R

d has been analyzed byDalang in [8]. Many other authors have also studied existence and uniqueness of solutionsto d-dimensional spatial stochastic equations, in particular, wave and heat equations, see[7, 9, 19, 27, 30] and references therein. Nualart et al. studied the probability densities ofthe mild solutions of the SPDEs with such a noise, using tools of the Malliavin calculus in[28, 29]. Under the assumption (C) and (Hη) for some η ∈ (0,1), Márquez-Carreras andSarrà [26] proved that the trajectories of the process uε(t, x) are (γ1, γ2)-Hölder continuouswith respect to the parameters t and x, satisfying γ1 ∈ (0, (1 − η)/2) and γ2 ∈ (0,1 − η).

As the parameter ε tends to zero, the solutions uε of (1) will tend to the solution of theequation defined by

⎧⎪⎨

⎪⎩

Lu0(t, x) = β(u0(t, x)), t ≥ 0, x ∈ [0,1]d ,u0(t, x) = 0, x ∈ ∂([0,1]d),u0(0, x) = 0, x ∈ [0,1]d .

(7)

In this paper we shall investigate deviations of uε from the deterministic solution u0, asε decreases to 0, that is, the asymptotic behavior of the trajectories,

Uε(t, x) := 1√εh(ε)

(uε − u0

)(t, x), (t, x) ∈ [0, T ] × [0,1]d ,

62 Y. Li et al.

where h(ε) is some deviation scale, which strongly influences the asymptotic behaviorof Uε .

The case h(ε) = 1/√

ε provides some large deviations estimates. Under the assumption(C) and (Hη) for some η ∈ (0,1), Márquez-Carreras and Sarrà in [26] proved that the lawof the solution uε satisfies a large deviation principle on the Hölder space, see Theorem 2.1for details.

If h(ε) is identically equal to 1, we are in the domain of the central limit theorem. Wewill show that (uε − u0)/

√ε converges as ε ↓ 0 to a random field.

To fill in the gap between the central limit theorem scale [h(ε) = 1] and the large devi-ations scale [h(ε) = 1/

√ε], we will study moderate deviations, that is when the deviation

scale satisfies

h(ε) → +∞,√

εh(ε) → 0 as ε → 0. (8)

The moderate deviation principle (MDP) enables us to refine the estimates obtained throughthe central limit theorem. It provides the asymptotic behavior for P(‖uε − u0‖ ≥ δ

√εh(ε))

while the central limit theorem gives asymptotic bounds for P(‖uε −u0‖ ≥ δ√

ε). Through-out this paper, we assume (8) is in place.

To prove the central limit theorem and the moderate deviation principle for the SPDE (1),we shall establish some exponential estimates in the space of Hölder continuous functions.The Garsia-Rodemich-Rumsey’s lemma will play a very important role.

The rest of this paper is organized as follows. In Sect. 2, the precise results are stated.In Sect. 3, we shall prove some convergence results of solutions. We establish the CLTand MDP in Sect. 4. We give some precise estimates of the fundamental solution G in theAppendix.

We conclude this section by introducing some basic notations and conventions used inthe paper. For any function φ : Dd

T → R, t ∈ [0, T ], let

|φ|t,∞ = sup{|φ(s, x)| : (s, x) ∈ [0, t] × [0,1]d},

‖φ‖t,γ1,γ2 = sup

{ |φ(s1, x1) − φ(s2, x2)||s1 − s2|γ1 + |x1 − x2|γ2

: (s1, x1) = (s2, x2) ∈ [0, t] × [0,1]d}

,

and let

‖|φ‖|γ1,γ2 = |φ|T ,∞ + ‖φ‖T ,γ1,γ2 .

Let Cγ1,γ2(DdT ;R) be the set of functions φ : Dd

T → R such that ‖|φ‖|γ1,γ2 < +∞, endowedwith the ‖| · ‖|γ1,γ2 -norm.

Throughout the paper, C(p) is a positive constant depending on the parameter p, andC is a constant depending on no specific parameter (except T and the Lipschitz constants),whose value may be different from line to line by convention.

2 Main Results

Let E be the space of all measurable functions ϕ :Rd → R such that

∫

Rd

dy

∫

Rd

dz|ϕ(y)|f (y − z)|ϕ(z)| < +∞,


endowed with the inner product

〈ϕ,ψ〉E :=∫

Rd

dy

∫

Rd

dzϕ(y)f (y − z)ψ(z).

Let H be the completion of (E, 〈·, ·〉E). For T > 0, let HT := L2([0, T ];H), which is a realseparable Hilbert space such that, if ϕ,ψ ∈ HT ,

E(F(ϕ)F (ψ)

) =∫ T

0

⟨ϕ(s, ·),ψ(s, ·)⟩Hds =: 〈ϕ,ψ〉HT

, (9)

where F is the noise introduced in Sect. 1.For any (t, x), (t ′, x ′) ∈ Dd

T , let

Γ((t, x),

(t ′, x ′)) :=

∫ t∧t ′

0ds

∫

R(x)

dy

∫

R(x′)dzf (y − z),

where R(x) is the rectangle [0, x] (here [0, x] is a product of the rectangles).By Márquez-Carreras and Sarrà [26], for any γ ∈ [0, 1

2 ), there exists an abstract Wienermeasure ν on Cγ,γ (Dd

T ;R) such that

∫

Cγ,γ (DdT

;R)

w(t, x)w(t ′, x ′)dν(ω) = Γ

((t, x),

(t ′, x ′)).

For any (t, x) ∈ DdT , set

WF (t, x) =∫ t

0

∫

R(x)

F (ds, dy).

Then WF is a Gaussian process with covariance function Γ . Its trajectories belong toCγ,γ (Dd

T ;R), and the law of the process WF is ν. Let the space H be the set of the functionsh such that, for any h ∈ HT , (t, x) ∈ Dd

T ,

h(t, x) = 〈1[0,t]×R(x), h〉HT,

with the scalar product

〈h, k〉H = 〈h, k〉HT.

Classical results on Gaussian processes (see [11, Theorem 3.4.12]) show that, for anyγ ∈ [0, 1

2 ), the family {εWF , ε > 0} satisfies a large deviation principle on Cγ,γ (DdT ;R) with

the rate function

I (h) ={

12‖h‖H , if h ∈ H ;+∞, otherwise.

(10)

For any h ∈ H,(t, x) ∈ DdT , consider the solution to the deterministic evolution equation

Sh(t, x) =∫ t

0ds

∫∫

[0,1]2d

dydzGt−s(x, y)α(Sh(s, y)

)f (y − z)h(s, z)

+∫ t

0ds

∫

[0,1]ddyGt−s(x, y)β

(Sh(s, y)

). (11)

64 Y. Li et al.

Recall the following theorem from Márquez-Carreras and Sarrà [26, Theorem 5.2]:

Theorem 2.1 Assume (C) and (Hη) for some η ∈ (0,1). Then the law of the solution uε

of Eq. (1) satisfies on Cγ,γ (DdT ;R), γ ∈ (0, (1 − η)/4), a large deviation principle with the

good rate function

S(g) ={

inf{I (h);Sh = g}, if g ∈ Im(S ·),+∞, otherwise.

(12)

More precisely, for any Borel measurable subset A of Cγ,γ (DdT ;R),

− infg∈Ao

S(g) ≤ lim infε→0

ε logP(uε ∈ A

)

≤ lim supε→0

ε logP(uε ∈ A

) ≤ − infg∈A

S(g),

where Ao and A denote the interior and the closure of A, respectively.

We furthermore suppose that

(D) the function β is differentiable, and its derivative β ′ is Lipschitz.

More precisely, there exists a positive constant K ′ such that

|β ′(y) − β ′(z)| ≤ K ′|y − z| for all y, z ∈R. (13)

Combined with the Lipschitz continuity of β , we conclude that

|β ′(z)| ≤ K, ∀z ∈R. (14)

Our first result is the following central limit theorem.

Theorem 2.2 Assume (C), (D) and (Hη) for some η ∈ (0,1). Then for any γ ∈ (0, (1 −η)/2) and r ≥ 1, the random field (uε − u0)/

√ε converges in Lr to a random field Y on

Cγ,2γ (DdT ;R), determined by

⎧⎪⎨

⎪⎩

LY (t, x) = α(u0(t, x))F (t, x) + β ′(u0(t, x))Y (t, x),

Y (t, x) = 0, x ∈ ∂([0,1]d),Y (0, x) = 0, x ∈ [0,1]d ,

(15)

for all (t, x) ∈ [0, T ] × [0,1]d .

Under conditions (C), (D) and (Hη) for some η ∈ (0,1), by Theorem 2.1 and in viewof the assumption (8), we see that, for any γ ∈ (0, (1 − η)/4), Y/h(ε) obeys an LDP onCγ,γ (Dd

T ;R) with the speed h2(ε) and with the good rate function

I (g) ={

inf{I (h);ShY = g}, if g ∈ Im(S ·

Y ),

+∞, otherwise,(16)


where ShY is the solution to the deterministic evolution equation

ShY (t, x) =

∫ t

0ds

∫∫

[0,1]2d

dydzGt−s(x, y)α(u0(s, y)

)f (y − z)h(s, z)

+∫ t

0ds

∫

[0,1]ddyGt−s(x, y)β ′(u0(s, y)

)Sh

Y (s, y), (17)

for any fixed h ∈ H,(t, x) ∈ DdT .

Our second main result reads as follows:

Theorem 2.3 Assume (C), (D) and (Hη) for some η ∈ (0,1). Then for any γ ∈ (0, (1 −η)/4), (uε − u0)/(

√εh(ε)) obeys an LDP on the space Cγ,γ (Dd

T ;R) with speed h2(ε) andwith rate function I given by (16).

3 Convergence of Solutions

The following estimate can be found in Márquez-Carreras and Sarrà [26, Proposition 3.1].

Lemma 3.1 Assume (C) and (H1). Then (1) has a unique solution. Moreover, for any T > 0,p ∈ [1,∞),

sup0<ε≤1

sup(t,x)∈Dd

T

E|uε(t, x)|p < ∞. (18)

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

Proposition 3.2 Assume (C) and (H1). Then for any p ≥ 2, there exists some constantC(p,K,T ) depending on p,K,T such that

E(|uε − u0|T ,∞

)p ≤ εp/2C(p,K,T ) → 0 as ε → 0. (19)

Proof Since

uε(t, x) − u0(t, x) =∫ t

0ds

∫

[0,1]ddyGt−s(x, y)

[β(uε(s, y)

) − β(u0(s, y)

)]

+ √ε

∫ t

0

∫


(uε(s, y)

)F(dsdy),

we have

(|uε − u0|T ,∞)p ≤ 2p−1

(

sup(t,x)∈Dd

T

∣∣∣∣

∫ t

0ds

∫


[β(uε(s, y)

) − β(u0(s, y)

)]∣∣∣∣

)p

+ εp2 2p−1

(

sup(t,x)∈Dd

T

∣∣∣∣

∫ t

0

∫


(uε(s, y)

)F(dsdy)

∣∣∣∣

)p

. (20)

Set

Mε1 (t, x) :=

∫ t

0ds

∫


[β(uε(s, y)

) − β(u0(s, y)

)],

66 Y. Li et al.

Mε2 (t, x) :=

∫ t

0

∫


(uε(s, y)

)F(dsdy).

By the Lipschitz condition (C), Hölder’s inequality and (58), for any p > (d + 2)/2, weobtain that

E(|Mε

1 |T ,∞)p ≤ Kp

(

sup(t,x)∈Dd

T

∫ t

0ds

∫

[0,1]ddyGq

s (x, y)

) pq

×E

∫ T

0

(|uε − u0|T ,∞)p

dt

≤ C(p,K,T )E

∫ T

0

(|uε − u0|T ,∞)p

dt, (21)

where 1/q + 1/p = 1.For any p > 2, x, y ∈ [0,1]d , 0 ≤ t ≤ T , by Burkholder’s inequality and Hölder’s in-

equality and (64), there exists γ1 ∈ (0,1 − η) such that

E∣∣Mε

2 (t, x) − Mε2

(t, x ′)∣∣p

≤ C(p)E

[∫ t

0ds

∫∫

[0,1]2d

dydz(∣∣Gt−s(x, y) − Gt−s

(x ′, y

)∣∣∣∣α

(uε(s, y)

)∣∣f (y − z)

× ∣∣Gt−s(x, z) − Gt−s

(x ′, z

)∣∣∣∣α

(uε(s, z)

)∣∣)] p

2

≤ C(p)

(∫ t

0ds

∫∫

[0,1]2d

dydz∣∣Gt−s(x, y) − Gt−s

(x ′, y

)∣∣f (y − z)

× ∣∣Gt−s(x, z) − Gt−s

(x ′, z

)∣∣

) p−22

∫ t

0ds

∫∫

[0,1]2d

dydz(∣∣Gt−s(x, y) − Gt−s

(x ′, y

)∣∣

× f (y − z)∣∣Gt−s(x, z) − Gt−s

(x ′, z

)∣∣E

[∣∣α

(uε(s, y)

)∣∣

p2∣∣α

(uε(s, z)

)∣∣

p2])

≤ C(p,K,T )(

1 + sup0<ε≤1

sup(t,x)∈Dd

T

E(|uε(t, x)|p))

×[∫ t

0ds

∫∫

[0,1]2d

dydz∣∣Gt−s(x, y) − Gt−s

(x ′, y

)∣∣f (y − z)

× ∣∣Gt−s(x, z) − Gt−s

(x ′, z

)∣∣

] p2

≤ C(p,K,T )(

1 + sup0<ε≤1

sup(t,x)∈Dd

T

E(|uε(t, x)|p)) · |x − x ′|γ1p. (22)

Similarly, in view of (55) and (56), it follows that for any 0 ≤ t ′ ≤ t ≤ T ,x ∈ [0,1]d andγ2 ∈ (0,

1−η

2 ),

E∣∣Mε

2 (t, x) − Mε2

(t ′, x

)∣∣p

≤ C(p,K,T )[1 + sup

0<ε≤1sup

(t,x)∈DdT

E(|uε(t, x)|p)]


×{(∫ t ′

0ds

∫∫

[0,1]2d

dydz|Gt−s(x, y) − Gt ′−s(x, y)|

× f (y − z)|Gt−s(x, z) − Gt ′−s(x, z)|) p

2

+(∫ t

t ′ds

∫∫

[0,1]2d

dydzGt−s(x, y)f (y − z)Gt−s(x, z)

) p2}

≤ C(p,K,T )[1 + sup

0<ε≤1sup

(t,x)∈DdT

E(|uε(t, x)|p)] · |t − t ′|γ2p. (23)

Putting together (22) and (23), we prove that for any (t, x), (t ′, x ′) ∈ DdT , γ ∈ (0,1 − η),

p ≥ 2,

E∣∣Mε

2 (t, x) − Mε2

(t ′, x ′)∣∣p ≤ C(p,K,T )

(|x − x ′| + |t − t ′|) pγ2 , (24)

where C(p,K,T ) is independent of ε. For p > 2(d + 1)/γ , applying Garsia-Rodemich-Rumsey’s lemma (Corollary 1.2 in [32]), there exist a random variable Np,ε(ω) and a con-stant C such that,

∣∣Mε

2 (t, x) − Mε2

(t ′, x ′)∣∣p

≤ Np,ε(ω)(|x − x ′| + |t − t ′|) pγ

2 −(d+1)

(

log

(C

|x − x ′| + |t − t ′|))2

, (25)

and

supε∈(0,1]

E[Np,ε] < +∞.

Choosing t ′ = 0 in (25), we obtain

E

(

sup(t,x)∈Dd

T

∫ t

0

∫


(uε(s, y)

)F(dsdy)

)p

≤ C(p,K,T ) supε∈(0,1]

E[Np,ε] < +∞. (26)

Putting (20), (21), (26) together, and using the Gronwall’s inequality, there exists a constantC(p,K,T ) such that

E(|uε − u0|T ,∞

)p ≤ εp/2C(p,K,T ) → 0, as ε → 0.

The proof is complete. �

4 The Proofs of Main Results

The following lemma is a consequence of the Garsia-Rodemich-Rumsey’s lemma, see[2, Lemma A1] for the proof.

Lemma 4.1 Let {V ε(t, x) : (t, x) ∈ [0, T ] × [0,1]d} be a family of real-valued stochasticprocesses and let p ∈ (1,∞). Assume

68 Y. Li et al.

(A1) For any (t, x) ∈ [0, T ] × [0,1]d ,

limε→0

E|V ε(t, x)|p = 0.

(A2) There exists γ0 > 0 such that for any (t, x), (t ′, x ′) ∈ [0, T ] × [0,1]d ,

E∣∣V ε(t, x) − V ε

(t ′, x ′)∣∣p ≤ C

(|t − t ′| + |x − x ′|2)d+1+γ0 ,

where C is a constant independent of ε.

Then for any γ ∈ (0, γ0/p), r ∈ [1,p),

limε→0

E(‖|V ε‖|γ,2γ

)r = 0.

4.1 Proof of Theorem 2.2

Proof Let Y ε = (uε − u0)/√

ε. We will prove that for any γ ∈ (0, (1 − η)/2), r ≥ 1,

limε→0

E(‖|Y ε − Y‖|γ,2γ

)r = 0. (27)

To this end, we will verify (A1), (A2) for V ε = Y ε − Y . Write

Y ε(t, x) − Y (t, x)

=∫ t

0

∫

[0,1]dGt−s(x, y)

(α(uε(s, y)

) − α(u0(s, y)

))F(dsdy)

+∫ t

0ds

∫


(β(uε(s, y)) − β(u0(s, y))√

ε− β ′(u0(s, y)

)Y (s, y)

)

= I ε1 (t, x) + I ε

2 (t, x) + I ε3 (t, x), (28)

where

I ε1 (t, x) :=

∫ t

0

∫

[0,1]dGt−s(x, y)

(α(uε(s, y)

) − α(u0(s, y)

))F(dsdy),

I ε2 (t, x) :=

∫ t

0ds

∫


[(β(uε(s, y)

) − β(u0(s, y)

))/√

ε

− β ′(u0(s, y))Y ε(s, y)

],

I ε3 (t, x) :=

∫ t

0ds

∫

[0,1]ddyGt−s(x, y)β ′(u0(s, y)

)(Y ε(s, y) − Y (s, y)

).

Now we shall divide the proof into the following two steps.

Step 1. Following the similar calculations in the proof of (26), and using (2), we deduce thatfor p > 2, t ∈ [0, T ],

E(|I ε

1 |t,∞)p ≤ C(p,K,T )E

(|uε − u0|t,∞)p ≤ ε

p2 C(p,K,T ). (29)


By Taylor’s formula, there exists a random field vε(t, x) taking values in (0,1) such that

β(uε(t, x)

) − β(u0(t, x)

) = β ′(u0(t, x) + vε(t, x)(uε(t, x) − u0(t, x)

))

× (uε(t, x) − u0(t, x)

).

Since β ′ is also Lipschitz continuous and vε(t, x) ∈ (0,1), we have

∣∣β ′(u0(t, x) + vε(t, x)

(uε(t, x) − u0(t, x)

)) − β ′(u0(t, x))∣∣

≤ K ′vε(t, x)∣∣uε(t, x) − u0(t, x)

∣∣

≤ K ′∣∣uε(t, x) − u0(t, x)∣∣. (30)

Hence

|I ε2 (t, x)| ≤ K ′

∫ t

0ds

∫


∣∣(uε(s, y) − u0(s, y)

)Y ε(s, y)

∣∣

= √εK ′

∫ t

0ds

∫

[0,1]ddyGt−s(x, y)|Y ε(s, y)|2.

By Hölder’s inequality, for any p > (d + 2)/2,

E(|I ε

2 |t,∞)p ≤ ε

p2 K ′p

(

sup(t,x)∈Dd

T

∣∣∣∣

∫ t

0ds

∫

[0,1]ddyG

qt−s(x, y)

∣∣∣∣

) pq

×∫ t

0E

(|Y ε|s,∞)2p

ds,

where 1/p+1/q = 1. Using (58) and Proposition 3.2, there exists a constant C(p,K,K ′, T )

depending on p,K,K ′, T such that

E(|I ε

2 |t,∞)p ≤ ε

p2 C

(p,K,K ′, T

). (31)

Noticing that |β ′| ≤ K , by Hölder’s inequality and (58), we deduce that for p > (d + 2)/2,

E(|I ε

3 |t,∞)p ≤ Kp

(

sup(t,x)∈Dd

T

∣∣∣∣

∫ t

0ds

∫

[0,1]ddyG

qt−s(x, y)

∣∣∣∣

) pq

∫ t

0E

(|Y ε − Y |s,∞)p

ds,

≤ C(p,K,T )

∫ t

0E

(|Y ε − Y |s,∞)p

ds, (32)

where 1/p + 1/q = 1. Putting (28), (29), (31), (32) together, we have

E(|Y ε − Y |t,∞

)p ≤ C(p,K,K ′, T

)(

εp2 +

∫ t

0E

(|Y ε − Y |s,∞)p

ds

)

.

By Gronwall’s inequality, we have

E(|Y ε − Y |T ,∞

)p ≤ εp2 C

(p,K,K ′, T

) → 0, as ε → 0, (33)

which in particular, implies (A1) in Lemma 4.1.

70 Y. Li et al.

Step 2. If the condition (A2) holds, that is there is a constant C independent of ε satisfyingthat for any γ0 ∈ (0,

1−η

2 ),

E∣∣(Y ε(t, x) − Y (t, x)

) − (Y ε

(t ′, x ′) − Y

(t ′, x ′))∣∣p ≤ C

(|t − t ′| + |x − x ′|2)pγ0 .

Then for any γ ∈ (0, γ0 − d+1p

), and r ∈ [1,p), by Lemma 4.1, we have

limε→0

E(‖|Y ε − Y‖|γ,2γ

)r = 0.

Since p ≥ d+22 ,0 < γ < γ0 − d+1

p<

1−η

2 is arbitrary, we get the desired result (27) for any

0 < γ <1−η

2 .Next, we will verify the condition (A2) for Y ε − Y . It suffices to show that all the terms

I εi , i = 1,2,3 satisfy (A2). The term I ε

1 is the most complicated one of the three, and theother two terms can be treated easier. So we only consider the first term. For any p > 2,x, x ′ ∈ [0,1]d , 0 ≤ t ≤ T ,γ0 ∈ (0, (1−η)/2), by Burkholder’s and Hölder’s inequalities, wehave

E∣∣I ε

1 (t, x) − I ε1

(t, x ′)∣∣p

≤ C(p)E

(∫ t

0

∥∥[Gt−s(x, ·) − Gt−s

(x ′, ·)][α(

uε(s, ·)) − α(u0(s, ·))]∥∥2

Hds

) p2

≤ C(p,K)

(∫ t

0

∥∥Gt−s(x, ·) − Gt−s

(x ′, ·)∥∥2

Hds

) p2 × sup

ε∈(0,1]E

(|uε − u0|T ,∞)p

≤ C(p,K)|x − x ′|2pγ0 , (34)

where (2), (64) and Proposition 3.2 were used. Similarly, in view of (55) and (56), it followsthat for 0 ≤ t ′ ≤ t ≤ T ,γ0 ∈ (0,

1−η

2 ),

E∣∣I ε

1 (t, x) − I ε1

(t ′, x

)∣∣p

≤ C(p)E

(∫ t ′

0

∥∥[Gt−s(x, ·) − Gt ′−s(x, ·)][α(

uε(s, ·)) − α(u0(s, ·))]∥∥2

Hds

) p2

+ C(p)E

(∫ t

t ′

∥∥Gt−s(x, ·)[α(

uε(s, ·)) − α(u0(s, ·))]∥∥2

Hds

) p2

≤ C(p,K)

(∫ t ′

0

∥∥Gt−s(x, ·) − Gt ′−s(x, ·)∥∥2

Hds

) p2 × sup

ε∈(0,1]E

(|uε − u0|T ,∞)p

+ C(p,K)

(∫ t

t ′

∥∥Gt−s(x, ·)∥∥2

Hds

) p2 × sup

ε∈(0,1]E

(|uε − u0|T ,∞)p

≤ C(p,K)|t − t ′|pγ0 , (35)

where Proposition 3.2 were used. The proof is complete. �

4.2 Proof of Theorem 2.3

The following lemma is a consequence of Márquez-Carreras and Sarrà [26, Lemma 6.2.1].


Lemma 4.2 Let Z : ([0, T ] × [0,1]d)2 → R, γ0 > 0 and CZ > 0 be such that for any (t, x)

and (t ′, x ′) ∈ [0, T ] × [0,1]d ,

∥∥Z(t, x, ·, ·) − Z

(t ′, x ′, ·, ·)∥∥2

HT≤ CZ

(|t − t ′| + |x − x ′|)2γ0 . (36)

Let N : Ω × [0, T ] × [0,1]d →R be an almost surely continuous, Ft -adapted process suchthat sup{|N(t, x)| : (t, x) ∈ [0, T ] × [0,1]d} ≤ ρ, a.s., and for (t, x) ∈ [0, T ] × [0,1]d , set

F(t, x) =∫ T

0

∫

[0,1]dZ(t, x, s, z)N(s, z)F (dsdz).

Then for all 0 < γ < γ0, there exist strictly positive constants L, C(γ, γ0) and C(γ,γ0) such

that for all M ≥ ρC12Z C(γ, γ0)C(γ, γ0),

P(‖|F‖|γ,γ ≥ M) ≤ L exp

(

− M2

ρ2CZC2(γ, γ0)

)

. (37)

Proof of Theorem 2.3 By the statements after Theorem 2.2 in Sect. 2, Y/h(ε) obeys an LDPon Cγ,γ (Dd

T ;R), with the speed function h2(ε) and the rate function I given by (16). Henceby [12, Theorem 4.2.13], to prove the LDP of Y ε/h(ε), it is enough to show that Y ε/h(ε)

is h2(ε)-exponentially equivalent to Y/h(ε), i.e., for any δ > 0,

lim supε→0

h−2(ε) logP

(‖|Y ε − Y‖|γ,γ

h(ε)> δ

)

= −∞. (38)

Since there exists a constant C(T ,d, γ ) satisfying that

‖|Y ε − Y‖|γ,γ ≤ C(T ,d, γ )‖Y ε − Y‖T ,γ,γ ,

to prove (38), it is enough to prove that

lim supε→0

h−2(ε) logP

(‖Y ε − Y‖T ,γ,γ

h(ε)> δ

)

= −∞, ∀δ > 0. (39)

Recall the decomposition in (28),

Y ε(t, x) − Y (t, x) = I ε1 (t, x) + I ε

2 (t, x) + I ε3 (t, x).

For any p > d+22 , q = p

p−1 , γ0 ∈ (0, 12 − d(q−1)

4 ), x, x ′ ∈ [0,1]d ,0 ≤ t ≤ T , by Hölder’sinequality, (14) and (65), we have

∣∣I ε

3 (t, x) − I ε3

(t, x ′)∣∣ ≤ K

(∫ t

0ds

∫

[0,1]ddy

∣∣Gt−s(x, y) − Gt−s

(x ′, y

)∣∣q

) 1q

×(∫ t

0ds

∫

[0,1]ddy|Y ε(s, y) − Y (s, y)|p

) 1p

≤ C(p,γ0,K,d,T )|x − x ′| 2γ0q ×

(∫ t

0

(|Y ε − Y |s,∞)p

ds

) 1p

. (40)

72 Y. Li et al.

Similarly, in view of (57) and (58), it follows that for 0 ≤ t ′ ≤ t ≤ T , p > (d +2)/2, q = p

p−1

and γ0 ∈ (0, 12 − d(q−1)

4 ),

∣∣I ε

3 (t, x) − I ε3

(t ′, x

)∣∣

≤ K

(∫ t ′

0ds

∫

[0,1]ddy|Gt−s(x, y) − Gt ′−s(x, y)|q

) 1q

×(∫ t ′

0ds

∫

[0,1]ddy|Y ε(s, y) − Y (s, y)|p

) 1p

+ K

(∫ t

t ′ds

∫

[0,1]ddy|Gt−s(y, z)|q

) 1q

×(∫ t

t ′ds

∫

[0,1]ddy|Y ε(s, y) − Y (s, y)|p

) 1p

≤ C(p,γ0,K,d,T )|t − s| 2γ0q ×

(∫ t

0

(|Y ε − Y |s,∞)p

ds

) 1p

. (41)

Putting together (40) and (41), we obtain that for p > d+22 , q = p

p−1 , γ0 ∈ (0, 12 − d(q−1)

4 ),0 ≤t ′ ≤ t ≤ T ,x, x ′ ∈ [0,1]d ,

∣∣I ε

3 (t, x) − I ε3

(t ′, x ′)∣∣ ≤ C(p,γ0,K,d,T )

(|t − t ′| + |x − x ′|)2γ0q

×(∫ t

0

(|Y ε − Y |s,∞)p

ds

) 1p

. (42)

Since p > d+22 and γ0 ∈ (0, 1

2 − d(q−1)

4 ) is arbitrary, we can choose p and γ0 satisfying thatγ = 2γ0/q for any γ ∈ (0,1/4). Noticing that |Y ε − Y |s,∞ ≤ (1 + s)γ ‖Y ε − Y‖s,γ,γ , weobtain that

‖I ε3 ‖t,γ,γ ≤ C

(∫ t

0

((1 + s)γ ‖Y ε − Y‖s,γ,γ

)pds

) 1p

.

Thus, for t ∈ [0, T ], we have

(‖Y ε − Y‖t,γ,γ

)p ≤ C(p,T ,K)

[(‖I ε

1 ‖t,γ,γ + ‖I ε2 ‖t,γ,γ

)p +∫ t

0

(‖Y ε − Y‖s,γ,γ

)pds

]

.

Applying Gronwall’s lemma to h(t) = (‖Y ε − Y‖t,γ,γ )p , we obtain that

(‖Y ε − Y‖T ,γ,γ

)p ≤C(p,T ,K)(‖I ε

1 ‖T ,γ,γ + ‖I ε2 ‖T ,γ,γ

)p. (43)

By (43), to prove (39), it is sufficient to prove that for any δ > 0

lim supε→0

h−2(ε) logP

(‖I εi ‖T ,γ,γ

h(ε)> δ

)

= −∞, i = 1,2.

Step 1. For any ε > 0 and θ > 0,

P(‖I ε

1 ‖T ,γ,γ > h(ε)δ) ≤ P

(‖I ε1 ‖T ,γ,γ > h(ε)δ, |uε − u0|T ,∞ < θ

) + P(|uε − u0|T ,∞ ≥ θ

).

(44)


By Lemmas 5.1 and 5.2, Gt−s(x, y)1[s≤t] satisfies (36) for γ0 = (1−η)/3. Applying Lemma4.2 with

Z(t, x, s, y) = Gt−s(x, y)1[s≤t], γ0 = (1 − η)/3, CZ = C, M = h(ε)δ, ρ = θK,

N(t, x) = (α(t, x, uε(t, x)

) − α(t, x, u0(t, x)

))1[|uε−u0|T ,∞<θ],

for any γ ∈ (0, (1 − η)/4) and for small ε such that h(ε)δ ≥ ρCC(γ, γ0)C(γ, γ0),

P(‖I ε

1 ‖T ,γ,γ > h(ε)δ, |uε − u0|T ,∞ < θ) ≤ L exp

(

− h2(ε)δ2

θ2K2CC2(γ, γ0)

)

. (45)

Since uε satisfies the large deviation principle on Cγ,γ (DdT ;R) (see Theorem 2.1),

lim supε→0

ε logP(|uε − u0|T ,∞ ≥ θ

) ≤ lim supε→0

ε logP(‖|uε − u0‖|γ,γ ≥ θ

)

≤ − inf{S(g) : ‖|g − u0‖|γ,γ ≥ θ

}.

where S is given by (12). Since the good rate function S(g) has compact level sets, theinf{S(g) : ‖|S(g) − u0‖|γ,γ ≥ θ} is obtained at some function g0. Because S(g) = 0 if andonly if g = u0, we conclude that

− inf{S(g) : ‖|g − u0‖|γ,γ ≥ θ

}< 0.

By (8), we have

lim supε→0

h−2(ε) logP(|uε − u0|T ,∞ ≥ θ

) = −∞. (46)

Since θ > 0 is arbitrary, putting together (44), (45) and (46), we obtain

lim supε→0

h−2(ε) logP

(‖I ε1 ‖T ,γ,γ

h(ε)> δ

)

= −∞. (47)

Step 2. For the second term,

I ε2 (t, x) =

∫ t

0ds

∫

[0,1]ddyGt−s(x, y)Bε(s, y),

where

Bε(s, y) := (

β(uε(s, y)

) − β(u0(s, y)

))/√

ε − β ′(u0(s, y))Y ε(s, y),

by Lemmas 5.1 and 5.2, following the same method in the proof of (42), we obtain that forany γ ∈ (0,1/4),

‖I ε2 ‖T ,γ,γ ≤ C(γ,T )|Bε|T ,∞. (48)

Similarly with (30), we have

|Bε|T ,∞ ≤ K ′(|uε − u0|T ,∞)2

/√

ε. (49)

By (2), (20) and (58) (taking p = 1), for any t ∈ [0, T ], we have

|uε − u0|t,∞ ≤ sup(l,x)∈Dd

t

∣∣∣∣

∫ l

0ds

∫

[0,1]ddyGl−s(x, y)K|uε(s, y) − u0(s, y)|

∣∣∣∣

74 Y. Li et al.

+ sup(l,x)∈Dd

t

∣∣∣∣√

ε

∫ l

0

∫

[0,1]dGl−s(x, y)α

(uε(s, y)

)F(dsdy)

∣∣∣∣

≤ C(K,T )

∫ t

0|uε − u0|s,∞ds + |I ε

2 |t,∞,

where

I ε2 (t, x) = √

ε

∫ t

0

∫


(uε(s, y)

)F(dsdy).

By the Gronwall’s inequality, we have

|uε − u0|T ,∞ ≤ C(K,T )|I ε2 |T ,∞ ≤ C(K,T )‖|I ε

2 ‖|γ,γ . (50)

Applying Lemma 4.2 with

Z(t, x, s, y) = Gt−s(x, y)1[s≤t], γ0 = (1 − η)/3, CZ = C, ρ = √εK

(1 + |u0|T ,∞ + θ

),

N(t, x) = √εα

(uε(t, x)

)1[|uε |T ,∞<|u0|T ,∞+θ],

for any fixed θ > 0, we obtain that for any γ ∈ (0, (1 − η)/4), M ≥ √εK(1 + |u0|T ,∞ +

θ)C12 C(γ,γ0)C(γ, γ0),

P(‖|I ε

2 ‖|γ,γ ≥ M, |uε|T ,∞ < |u0|T ,∞ + θ)

≤ L exp

(

− M2

εK2CC2(γ, γ0)(1 + |u0|T ,∞ + θ)2

)

. (51)

Similarly with (46), we have

lim supε→0

h−2(ε) logP(|uε|T ,∞ ≥ |u0|T ,∞ + θ

)

≤ lim supε→0

h−2(ε) logP(|uε − u0|T ,∞ ≥ θ

)

= −∞. (52)

Putting (49)–(52) together, we have

lim supε→0

h−2(ε) logP

(‖I ε2 ‖T ,γ,γ

h(ε)> δ

)

≤ lim supε→0

h−2(ε) logP

((‖|I ε

2 ‖|γ,γ

)2>

√εh(ε)δ

C(γ,T ,K)

)

≤ lim supε→0

h−2(ε) log

[

P

((‖|I ε

2 ‖|γ,γ

)2>

√εh(ε)δ

C(γ,T ,K), |uε|T ,∞ < |u0|T ,∞ + θ

)

+ P(|uε|T ,∞ ≥ |u0|T ,∞ + θ

)]

≤(

lim supε→0

−δ√εh(ε)C(γ,T ,K)K2CC2(γ, γ0)(1 + |u0|T ,∞ + θ)2

)


∨(

lim supε→0

h−2(ε) logP(|uε|T ,∞ ≥ |u0|T ,∞ + θ

))

= −∞.


Acknowledgements The authors are grateful to the anonymous referees for conscientious commentsand corrections. Y. Li and S. Zhang were supported by Natural Science Foundation of China (11471304,11401556). R. Wang was supported by Natural Science Foundation of China (11301498, 11431014).

Appendix

To make reading easier, we present here some results on the kernel G partially fromMárquez-Carreras and Sarrà [26].

Recall that G denotes the fundamental solution of the heat equation

⎧⎪⎨

⎪⎩

∂∂t

Gt (x, y) = �xGt(x, y), t ≥ 0, x, y ∈ [0,1]d ,Gt (x, y) = 0, x ∈ ∂([0,1]d),G0(x, y) = δ(x − y).

(53)

The details about the construction of the fundamental solution G can be found in [18,Chap. 1]. This fundamental solution G is non-negative and can be decomposed into differentterms as follows (see [15]):

Gt−s(x, y) = H(t − s, x − y) + R(t − s, x, y) +∑

i∈Id

Hi(t − s, x − y), (54)

where H is the heat kernel on Rd ,

H(t, x) =(

1

4πt

) d2

exp

(

−|x − y|24t

)

,

R is a Lipschitz function, Id = {i = (i1, . . . , id) ∈ {−1,0,1}d − {(0, . . . ,0)}} and

Hi(t − s, x − y) = (−1)kH(t − s, x − yi

),

with k = ∑d

j=1 |ij | and⎧⎪⎨

⎪⎩

yij = yj , if ij = 0,

yij = −yj , if ij = 1,

yij = 2 − yj , if ij = −1.

Lemma 5.1 Assume (Hη) for some η ∈ (0,1). There exists a positive constant C indepen-dent of t and x such that, for any 0 ≤ t ′ < t ≤ T ,x ∈ [0,1]d , γ1 ∈ (0,

1−η

2 ), 1 ≤ p < 1 + 2d

and γ2 ∈ [0, 12 − d(p−1)

4 ),

∫ t ′

0

∥∥Gt−s(x, ·) − Gt ′−s(x, ·)∥∥2

Hds ≤ C|t − t ′|2γ1 , (55)

76 Y. Li et al.

∫ t

t ′

∥∥Gt−s(x, ·)∥∥2

Hds ≤ C|t − t ′|2γ1 , (56)

∫ t ′

0ds

∫

[0,1]ddy|Gt−s(x, y) − Gt ′−s(x, y)|p ≤ C|t − t ′|2γ2 , (57)

∫ t

t ′ds

∫

[0,1]ddy|Gt−s(x, y)|p ≤ C|t − t ′|2γ2 . (58)

Proof The proof of (55) and (56) can be found in Lemma 6.1.3 in [26]. The proofs of (57)and (58) are similar. For the convenience of reading, we shall give the proofs of (57) and(58).

For any p ∈ [1,1 + 2d), (54) implies that

|Gt−s(x, y) − Gt ′−s(x, y)|p ≤ C(d,p)

(∣∣H(t − s, x − y) − H

(t ′ − s, x − y

)∣∣p

+ ∣∣R(t − s, x − y) − R

(t ′ − s, x − y

)∣∣p

+∑

i∈Id

∣∣Hi(t − s, x − y) − Hi

(t ′ − s, x − y

)∣∣p

)

. (59)

In order to check (57), we only need to bound the right-hand side of (59). Let L be theLipschitz constant of the function R. Clearly

∫ t ′

0ds

∫

[0,1]ddy

∣∣R(t − s, x, y) − R

(t ′ − s, x, y

)∣∣p ≤ LpT |t − t ′|p. (60)

The terms Hi can be studied using the same arguments as H . Now it remains to analyze theterm with H . First, for any p ∈ [1,1 + 2

d),

∫ t ′

0ds

∫

[0,1]ddy

∣∣H(t − s, x − y) − H

(t ′ − s, x − y

)∣∣p ≤ C(A1 + A2), (61)

with

A1 =∫ t ′

0ds

∫

[0,1]ddy

(1

4π(t − s)

) pd2

∣∣∣∣exp

(

−|x − y|24(t − s)

)

− exp

(

− |x − y|24(t ′ − s)

)∣∣∣∣

p

,

A2 =∫ t ′

0ds

∫

[0,1]ddy

∣∣∣∣

(1

4π(t − s)

) d2 −

(1

4π(t ′ − s)

) d2∣∣∣∣

p

exp

(

−p|x − y|24(t ′ − s)

)

.

Since t ′ < t , we have

exp

(

−|x − y|24(t − s)

)

≥ exp

(

− |x − y|24(t ′ − s)

)

.

Using the inequality

(a − b)p ≤ ap − bp for all a > b > 0, p ≥ 1,


we obtain that

A1 ≤∫ t ′

0ds

∫

Rd

dy

(1

4π(t − s)

) pd2

[

exp

(

−p|x − y|24(t − s)

)

− exp

(

−p|x − y|24(t ′ − s)

)]

=∫ t ′

0

(1

4π(t − s)

) d(p−1)2

p− d2

[

1 −(

t ′ − s

t − s

) d2]

.

Since for any γ2 ∈ [0, 12 − d(p−1)

4 ) and s < t ′,

t − t ′

t − s<

(t − t ′

t − s

)2γ2

≤ 1,

(t ′ − s

t − s

)n

≤ t ′ − s

t − s< 1, n ≥ 1,

we have

1 −(

t ′ − s

t − s

) d2 ≤

[

1 −(

t ′ − s

t − s

)]

×(

1 + t ′ − s

t − s+

(t ′ − s

t − s

)2

+ · · · +(

t ′ − s

t − s

)� d2 �)

≤(⌊

d

2

⌋

+ 1

)(t − t ′

t − s

)

≤(⌊

d

2

⌋

+ 1

)(t − t ′

t − s

)2γ2

,

where � d2 � is the largest integer number less or equal than d

2 .Thus, for any γ2 ∈ [0, 1

2 − d(p−1)

4 ), we have

A1 ≤ (4π)− d(p−1)2 p− d

2

(⌊d

2

⌋

+ 1

)

· |t − t ′|2γ2 ·∫ t ′

0ds

(t ′ − s

)− d(p−1)2 −2γ2

≤ C(p,γ2, d, T )|t − t ′|2γ2 . (62)

Similar computations to the study of A2, we have

A2 ≤∫ t ′

0ds

∫

Rd

dy

∣∣∣∣

(1

4π(t − s)

) d2 −

(1

4π(t ′ − s)

) d2∣∣∣∣

p

exp

(

−p|x − y|24(t ′ − s)

)

=∫ t ′

0ds

∣∣∣∣

(1

4π(t − s)

) d2 −

(1

4π(t ′ − s)

) d2∣∣∣∣

p

·(

4π(t ′ − s)

p

) d2

= (4π)− d(p−1)2 p− d

2

∫ t ′

0ds|(t − s)− d

2 − (t ′ − s

)− d2 |p−1 ·

(

1 −(

t ′ − s

t − s

) d2)

≤ (4π)− d(p−1)2 p− d

2

∫ t ′

0ds

(t ′ − s

)− d(p−1)2 ·

(

1 −(

t ′ − s

t − s

) d2)

≤ (4π)− d(p−1)2 p− d

2

(⌊d

2

⌋

+ 1

)

· |t − t ′|2γ2 ·∫ t ′

0ds

(t ′ − s

)− d(p−1)2 −2γ2

≤ C(p,d, γ2, T )|t − t ′|2γ2 . (63)

Then (57) follows form (59)–(63).

78 Y. Li et al.

Next we shall prove (58). As before, we only need to check (58) replacing G by H . Forany 0 ≤ t ′ ≤ t ≤ T , 1 ≤ p < 1 + 2

dand γ2 ∈ [0, 1

2 − d(p−1)

4 ), we have

∫ t

t ′ds

∫

[0,1]ddyH(t − s, x − y)p

≤∫ t

t ′ds

∫

Rd

dy

(1

4π(t − s)

) pd2

exp

(

−p|x − y|24(t − s)

)

= (4π)− (p−1)d2 p− d

2

∫ t

t ′ds(t − s)− (p−1)d

2

= (4π)− (p−1)d2 p− d

2

(

1 − (p − 1)d

2

)−1(t − t ′

)1− (p−1)d2

≤ C(p,γ2, d, T )|t − t ′|2γ2 .


Lemma 5.2 Assume (Hη) for some η ∈ (0,1). There exists a positive constant C indepen-dent of t and x such that, for any 0 ≤ t ≤ T ,x, x ′ ∈ [0,1]d , p ∈ [1,1 + 2

d), γ1 ∈ (0,1 − η)

and γ2 ∈ [0, 12 − d(p−1)

4 ),

∫ t

0

∥∥Gt−s(x, ·) − Gt−s

(x ′, ·)∥∥2

Hds ≤ C|x − x ′|2γ1 , (64)

∫ t

0ds

∫

[0,1]ddy

∣∣Gt−s(x, y) − Gt−s

(x ′, y

)∣∣p ≤ C|x − x ′|2γ2 . (65)

Proof The proof of (64) can be found in Lemma 6.1.4 in [26]. Now we shall give the proofof (65). Using the similar arguments as in the proof of (57), we only need to check (65)replacing G by H . For any x, x ′ ∈ [0,1]d , p ∈ [1,1 + 2

d) and γ2 ∈ [0, 1

2 − d(p−1)

4 ),

B :=∫ t

0ds

∫

[0,1]ddy

∣∣H(t − s, x − y) − H

(t − s, x ′ − y

)∣∣p

≤ C

∫ t

0ds

∫

Rd

dy(4π(t − s)

)− pd2

∣∣∣∣exp

(

−|x − y|24(t − s)

)

− exp

(

−|x ′ − y|24(t − s)

)∣∣∣∣

2γ2

×(

exp

(

− (p − 2γ2)|x − y|24(t − s)

)

+ exp

(

− (p − 2γ2)|x ′ − y|24(t − s)

))

.

By the mean-value theorem, and the fact e−x < 0 for any x > 0, we have

B ≤ C

∫ t

0ds

∫

Rd

dy(4π(t − s)

)− pd2

( |x − x ′|t − s

)2γ2

×(

exp

(

− (p − 2γ2)|x − y|24(t − s)

)

+ exp

(

− (p − 2γ2)|x ′ − y|24(t − s)

))

≤ 2C(4π)− d(p−1)2 (p − 2γ2)

− d2 · |x − x ′|2γ2 ·

∫ t

0ds(t − s)− d(p−1)

2 −2γ2

≤ C(p,γ2, d, T )|x − x ′|2γ2 .



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