Research ArticleModerate Deviations for Stochastic Fractional Heat EquationDriven by Fractional Noise

Xichao Sun ,1 Ming Li ,2 and Wei Zhao3

1College of Science, Bengbu University, 1866 Caoshan Rd., Bengbu 233030, China2School of Information Science and Technology, East China Normal University, No. 500, Dong-Chuan Road, Shanghai 200241, China3American University of Sharjah, P.O. Box 26666, Sharjah, UAE

Correspondence should be addressed to Xichao Sun; sunxichao626@126.com

Received 28 January 2018; Revised 6 April 2018; Accepted 2 May 2018; Published 10 July 2018

Academic Editor: Maricel Agop

Copyright © 2018 Xichao Sun et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider a class of stochastic fractional heat equations driven by fractional noises. A central limit theorem is given, and amoderate deviation principle is established.

1. Introduction

Since the work of Freidlin and Wentzell [1], the largedeviation principle (LDP) has been extensively developedfor small noise systems and other types of models (suchas interacting particle systems) (see [2–7]). Cardon-Weber[2] proved a LDP for a Burgers-type SPDE driven by whitenoise. Marquez-Carreras and Sarra [3] proved a LDP for astochastic heat equation with spatially correlated noise, andMellali and Mellouk [4] extended Marquez-Carreras andSarra’s [3] to a fractional operator. Jiang et al. [5] proveda LDP for a fourth-order stochastic heat equation drivenby fractional noise. Budhiraja et al. [6] studied large devia-tion properties of systems of weakly interacting particles.Budhiraja et al. [7] proved a large deviation for Brownianparticle systems with killing.

Similar to the large deviation, the moderate deviationproblems also come from the theory of statistical inference.Using the moderate deviation principle (MDP), we canget the rate of convergence and an important method toconstruct asymptotic confidence intervals, for example,Liming [8], Guillin and Liptser [9], Cattani and Ciancio[10], and other references therein. There are also manyworks about MDP about stochastic (partial) differentialequations; some surveys and literatures could be foundin Budhiraja et al. [11], Wang and Zhang [12], Li et al.[13], Yang and Jiang [14], and the references therein. On

the other hand, fractional equations have attracted manyphysicists and mathematicians due to various applicationsin risk management, image analysis, and statistical mechan-ics (see Droniou and Imbert [15], Bakhoum and Toma [16],Levy and Pinchas [17], Mardani et al. [18], Niculescu et al.[19], Paun [20], and Pinchas [21] for a survey of applica-tions). Stochastic partial differential equations involving afractional Laplacian operator have been studied by manyauthors; see Mueller [22], Wu [23], Liu et al. [24], Wu[25], and the references therein.

Motived above, we investigated the moderate deviationsabout the stochastic fractional heat equation with fractionalnoise as follows:

∂vε

∂tt, x =Dδ,αvε t, x + f vε t, x + εBH dt, dx ,

vε o, x = 0,1

where t ∈ 0, T , x ∈D = 0, 1 , Dδ,α is the fractional Laplacianoperator which is defined in Appendix, and BH dt, dxdenotes a fractional noise which is fractional in time andwhite in space with Hurst parameter H ∈ 1/2, 1 ; thatis, BH is a mean zero Gaussian random field on 0, T ×Dwith covariance.

HindawiComplexityVolume 2018, Article ID 7402764, 17 pageshttps://doi.org/10.1155/2018/7402764

http://orcid.org/0000-0002-4363-6397http://orcid.org/0000-0002-2725-353Xhttps://doi.org/10.1155/2018/7402764

Cov BH t, x BH s, y =12

t2H + s2H − t − s 2H x∧y

2

Assume that the coefficients satisfy the following.

Assumption 1. Function f is Lipschitz; that is, there exist anm > 0 satisfying

f y − f x ≤m y − x , ∀x, y ∈D 3

Under the conditions of Assumption 1, (1) possesses aunique solution in the sense of Walsh [26] as follows:

vε t, x =t

0 DGδα t − s, x − z f v

ε s, z dzds

+ εt

0 DGδα t − s, x − z B

H ds, dz4

As the parameter ε→ 0, the solution vε t, x of(1) will tend to v0 t, x which is the solution to the fol-lowing equation:

∂v0

∂tt, x =Dδ,αv0 t, x + f v0 t, x ,

v0 0, x = 05

This paper mainly devotes to investigate the deviations ofvε from the deterministic solution v0, as ε→ 0, that is, theasymptotic behavior of the trajectories.

Vε t, x ≔1εa ε

vε − v0 t, x ,  t, x ∈ 0, T ×D, 6

where a ε is the same deviation scale that strongly influ-ences the asymptotic behavior of Vε.

If a ε = 1/ ε, we are in the domain of large deviationestimate, which can be proved similarly to Jiang et al. [5].

The case a ε ≡ 1 provides the central limit theorem. Asε↓0, we will prove that vε − v0 / ε converges to a randomfield in this paper.

To fill the gap between scale a ε = 1 and scale a ε =1/ ε, we mainly devote to the moderate deviation whenthe scale satisfies the following:

a ε → +∞,  εa ε → 0, ε→ 0 7

This paper is organized as follows. In Section 2, thedefinition of the fractional noise BH ds, dz is given. InSection 3, the main result is given and proved. In Appendix,some results about the Green kernel are given.

2. Fractional Noise

Let H ∈ 1/2, 1 , and BH 0, t × A t,A ∈ 0,T ×B ℝ is acentered Gaussian family of random variables with thecovariance satisfying

E BH t, A BH s, B = A ∩ B RH t, s , 8

with s, t ∈ 0, T , A, B ∈B ℝ and covariance kernel RHt, s = 1/2 t2H + s2H − t − s 2H , where ∣A∣ denotes theLebesgue measure of the set A ∈B ℝ and B ℝ denotesthe class of Borel sets in ℝ.

We denote φ as the set of step functions on 0, T ×ℝ. LetH be the Hilbert space defined as the closure of φ withrespect to the scalar product.

1 0,t ×A, 1 0,s ×BH= E BH t, A BH s, B 9

According to Nualart and Ouknine [27], the mapping1 0,t ×A → BH t, A can be extended to an isometry betweenH and the Gaussian space H1 B

H associated with BH anddenoted by

ϕ↦BH ϕ ≔t

0 Aϕ s, x BH ds, dx 10

Define the linear operator K∗H φ↦L2 0, T by

K∗H ϕ = KH T , s ϕ s, y +T

sϕ r, y − ϕ s, y

∂KH∂r

r, s dr,

11

where KH is defined by

with CH = 2αΓ 3/2 − α /Γ α + 1/2 Γ 2 − 2α 1/2, andone can get

∂KH∂t

t, s = CH12−H t − s H− 3/2

ts

H− 1/213

Moreover, KH satisfies the following:

s∧t

0KH s, r KH t, r dr =

12

t2H + s2H − t − s 2H 14

KH t, s ≔CHs

1/2 −Ht

su − s H− 3/2 1 − s

u

1/2 −Hdu + CH t − s

H− 1/2 , 0 < s ≤ t,

0, otherwise,12

2 Complexity

Then, since

K∗H1 0,t ×A s, x = KH t, s 1 0,t ×A s, x , 15

one can get

K∗H ϕ , K∗H ψ L2 0,T ×D = ϕ, ψ ℋ, 16

where ϕ and ψ in φ are any step functions. So theoperator K∗H gives an isometry between the Hilbert spaceℋ and L2 0, T × 0, 1 . Hence, W t, A , t ∈ 0, T ,A ∈ℬ 0, 1 defined by

W t, A = BH K∗H−1 1 0,t ×A 17

is a space-time white noise, and BH has the followingform:

BH t, x =t

0

x

0KH t, s W ds, dy 18

Therefore, the mild formulation of (4) has the followingform:

vε t, x =t

0 DGδα t − s, x − z f v

ε s, z dzds

+ εt

0 DK∗HG

δα t − s, x − z W ds, dz

19

That is, the last term of (4) is equal to

t

0 DGδα t − s, x − z B

H ds, dz

=t

0 DK∗HG

δα t − s, x − z W ds, dz

20

The following embedding proposition is given by Nualartand Ouknine [27].

Lemma 1. Set H > 1/2 , then, we have

L1/H 0, T ×ℝ ⊂ℋ 21

3. Main Results and Their Proof

3.1. Main Results. For any function ϕ defined on 0, T ×D,let

where μ ∈ 0, α + 1 H − 1/α , θ ∈ 0, min 1, α + 1 H − 1 ,and γ ∈ 0, 1 . Let

ϕ γ = ϕ T ,∞ + ϕ T ,γ 23

Let Cγ 0, T ×D be the functions ϕ 0, T ×D whichsatisfy ϕ γ

Theorem 1. Let H ∈ 1/2, 1 . Under Assumption1, the lawof the solution to (1) satisfies a deviation principle onCγ D × 0, T with the good rate function:

I φ = inf12

e 2ℰ ; Ze = φ , 30

with the convention inf∅ =∞, where 0 < γ < 1.More precisely, for any Borel measurable subset B of

Cγ 0, T ×D ,

−infφ∈Bo I φ ≤ lim infε→0 ε log P Ze ∈ B

≤ lim supε→0

ε log P Ze ∈ B ≤ −infφ∈BI φ ,31

where Bo and B denote the interior and the closure of B,respectively.

We furthermore suppose that the coefficients satisfy thefollowing.

Assumption 2. f is differentiable, and the derivative f ′ of f isLipschitz. That is to say, there exist positive constant m andm′ which satisfy the following:

f ′ y − f ′ x ≤m′ y − x , ∀x, y ∈D 32

Together with the Lipschitz of f , we conclude that

f ′ x ≤m 33

Now, we give the following central limit theorem.

Theorem 2. Let f and its derivative f ′ satisfy Assumptions 1and 2. Then, for p ≥ 1, ve − v0 / ε converges in Lp to a ran-dom field U on Cγ 0, T ×D with 0 < γ < 1, determined by∂U t, x

∂t=Dδ,αU t, x + f ′ v0 t, x U t, x + BH dt, dx ,

U 0, x = 0,34

for all, t, x ∈ 0, T ×D.

Let the function Ue be the solution to the followingpartial differential equation:

∂Ue t, x∂t

=Dδ,αUe t, x + f ′ v0 t, x Ue t, x + F e t, x ,

Ue 0, x = 035

Under Assumptions 1 and 2, by Theorem 1, one canget U/a ε which satisfies large deviation principles onCγ 0, T ×D with the speed e2 ε and the good ratefunction satisfies the following:

I φinf

12

e 2ℰ, Ue = φ ,

+∞, otherwise36

Now, the second result is given as follows:

Theorem 3. In moderate deviation principle, let H ∈ 1/2, 1 .Under the Assumptions1and2, then, the random field 1/εa ε vε − v0 satisfies a large deviation principle on

the space Cγ 0, T ×D with speed a2 ε and the goodrate function I φ defined by (36), where 0 < γ < 1.

3.2. Convergence of the Solution

Lemma 2. Let H ∈ 1/2, 1 . Under Assumption1, then,there exists a unique solution to (1). Moreover, for anyp ∈ 1,∞ , T > 0,

supε≥1

supt,x ∈ 0,T ×D

E vε t, x p

Set

Nε1 t, x =t

0 DGδα t − s, x − z f v

ε s, z − f v0 s, z dzds,

Nε2 t, x =t

0 DGδα t − s, x − z B

H dsdz

41Together with Hölder’s inequality, the Lipschitz condi-

tion (C) and (5) of Lemma A.1, for 1/α < q < α, we have

E Nε1 T ,∞p ≤mp sup

t,x ∈ 0,T ×DGδα t − s, x − z

qdzds

p/q

× ET

0vε − v0

T ,∞

pdt

≤ C p,m, T ET

0vε − v0

T ,∞

pdt,

42where 1/p + 1/q = 1.

For any 0 ≤ t ≤ T , x, z ∈D, p > 2, by (21) and (A.9),there exist θ ∈ 0, min 1, α + 1 H − 1 which satisfiesthe following:

E Nε2 t, x −Nε2 t, x′

p

= Et

0 DGδα t − s, x − z − G

δα t − s, x′ − z BH dsdz

p

≤t

0 DK∗H G

δα t − s, x − z − G

δα t − s, x′ − z

2dsdz

p/2

≤t

0 DGδα t − s, x − z −G

δα t − s, x′ − z

1/Hdsdz

Hp

≤m x − x′ pθ

43

Similarly,

By (21), (A.10), and (A.11), for μ ∈ 0, α + 1 H − 1 /α , onecan get

t

t DK∗H G

δα t − s, x − z

2dsdz

p/2

≤ CHt

t DGδα t − s, x − z

1/Hdsdz

Hp

≤ CH t − t′p α+1 H−1 /α

45

and

t

0 DK∗H G

δα t − s, x − z − G

δα t′ − s, x − z

2dsdz

p/2

≤ CHt

t DK∗H G

δα t − s, x − z −G

δα t′ − s, x − z

1/Hdsdz

Hp

≤ CH t − t′pμ

46

Together with (43), (44), (45), and (46), one can get forany p ≥ 2 and t′, x′ , t, x ∈ 0, T ×D, there exists aconstant β > 0 such that

E Nε2 t, x −Nε2 t, x′

p≤ C p,m,H, T x − x′ θ + t − t′ μ

pβ,

47

where C p,m,H, T is independent of ε. For p > 2/β, byGarsis-Rodemich-Rumser’s Lemma, there exist a constantC and a random variable Mp,ε ω satisfying

Nε2 t, x −Nε2 t, x′

p

≤mp,ε ω x − x′θ + t − t′ μ

pβ−2

⋅ logc

x − x θ + t − t μ

248

E Nε2 t, x −Nε2 t′, x

p= E

t

0 DGδα t − s, x − z − G

δα t′ − s, x − z BH dsdz

p

≤t

0 DK∗H G

δα t − s, x − z −G

δα t′ − s, x − z

2dsdz

p/2

≤ C p,m, Tt

t DK∗H G

δα t − s, x − z

2dsdz

p/2

+ C p,m, Tt

0 DK∗H G

δα t − s, x − z −G

δα t − s, x − z

2dsdz

p/2

44

5Complexity

and

supε∈ 0,1

E mp,ε 0 and ϵ0 > 0satisfying I1 ≤ a and ϵ ≤ ϵ0 for all h ∈ E1, and

P d1 Xϵ1 ≤ ρ, d2 x

ϵ2, K e ≥ δ ≤ exp −

Rϵ2

52

Then, Xϵ2, ϵ > 0 satisfies a LDP with the rate functions

I h = inf I1 e : K e = h 53

To prove Theorem 1, one only needs to prove

(i) Under some topology, Z e : e e E ≤ a →C 0, T , Lp D is continuous for any a > 0.

(ii) In Freidin-Wentzell inequality, for any R > 0, η > 0,and e ∈E, there exist a δ > 0 satisfying

Theorem 4. When the level set e E ≤ a is endowed withthe topology of uniform convergence on 0, T ×D ,

Z e : e eE≤ a → C 0, T , Lp D 55

is the continuous map for any a ∈ 0, +∞ .

Proof. One only needs to prove that for fixed a > 0, e, h ∈∥e∥E ≤ a ,

supt∈ 0,T ℝ

Z e t, x − Z h t, x 2dx ≤ C e − h ℰ 56

Note that

Using (A.5) in Appendix with p = q = 2, then ρ = 1, onecan get

limϵ→0

sup ϵ2logP ϵW − e ∞ < δ, supt∈ 0,T

uϵ t, · − Z e t, · ∥2 ≥ ρ ≤ −R 54

Z e t, x − Z h t, x =t

0 ℝGδα t − s, x − z f s, z, Z e s, z − f s, z, Z h s, z dzds

+t

0 ℝK∗HG

δα t − s, x − z e s, z dzds≔ I t, x + II t, x

57

I t, x 22 ≤ Ct

0t − s − 1/α + 1/αρ f s, z, Z e s, z − f s, z, Z h s, z 2ds ≤ C sup

0≤s≤tZ e s, ⋅ − Z h s, ⋅ 2 58

6 Complexity

Now, we deal with II t, x , together with

Recalling K∗H is defined by (11), one can get

K∗Hϕ s, x ≤ maxw∈ s,T ϕ w, x KH T , s ,

M t =ℝ

t

0 ℝK∗HG

δα t − s, x − z

2dzdsdx

≤ℝ

t

0 ℝmaxw∈ s,t

Gδα t − s, x − z2K2H t, s dzdxds

≤ Ct

0t − s − 1/α t − s 2H−1ds 0, we define

W t, x =W t, x −e x, tϵ 62

and

dpdp

= exp1ϵ

T

0 ℝe t, x W dt, dx −

12ϵ2

T

0 ℝe2 t, x dxdt ≔ ZT

63

Using Girsanov’s theorem, the process W is a Browniansheet under P. Suppose vε t, x is a solution of (1) underp. Then,

vϵ t, x =ℝGδα t, x − z v0 z dz

+t

0 ℝGδα t − s, x − z f t, x, v

ε s, z dzds

+t

0 ℝK∗HG

δα t − s, x − z e s, z dzds

+ ϵt

0 ℝK∗HG

δα t − s, x − z W ds, dz

64

Now, one can prove (36). Note that, under p, then,

So under p, by Gronwall’s Lemma, one can get

II t, x 22 =ℝ

t

0 ℝK∗HG

δα t − s, x − z e s, z − h s, z dzds

2dx

≤ Cℝ

t

0 ℝK∗HG

δα t − s, x − z

2dzdsdx × e − h 2E ≔ CM t × e − h

2ℰ

59

uϵ t, · − Z e t, · 2 =t

0 ℝGδα t − s, · − z f t, x, u

ε s, z − f t, x, Z e s, z dzds2

+ ϵt

0 ℝK∗HG

δα t − s, · − y W ds, dz

2

≤t

0 ℝuε s, · − Z e s, · 2dzds + ϵ

t

0 ℝK∗HG

δα t − s, · − z W ds, dz

2

65

supt∈ 0,T

uε s, · − Z h s, · 2 ≤ C supt∈ 0,T

ϵt

0 ℝK∗HG

δα t − s, x − z W ds, dz

266

7Complexity

Now, one can change (34) the proof to the followingtheorem.

Theorem 5. Suppose e ∈E and e ∈E ≤ a. For each R > 0,η > 0, and e ∈E, there exists a constant δ > 0 satisfying

In the following, we give a key Lemma to prove Theorem1, which is similar to Candon-Weber [2], and the proof isomitted.

Lemma 3. Suppose F 0, T ×D 2 →ℝ, CF > 0 and β0 > 0satisfying

T

0 DF t, x, v, z − F s, y, v, z 2dudz ≤ CF t − s

μ + x − y θβ0,

68

for any s, z and t, x ∈ 0, T ×D. Suppose N Ω ×0, T × 0, 1 →ℝ which is an almost surely continuous,F t-adapted process satisfying sup N t, x : t, x ∈ 0, T ×D ≤ ρ, a, s, and for t, x ∈ 0, T ×D, suppose

F t, x =T

0 DF t, x, v, z N v, z W dvdz 69

Then, for any 0 < β < β0/2 , there exist a positive constantC, Ç(β, β0), and C β, β0 such that for allM ≥ ρC

1/2F Ç(β, β0)

C β, β0 ,

P F t,β ≥M ≤ C exp −M2

ρ2CzCβ, β0 C2 β, β0

70

Proof of Theorem 5. Suppose

F t, x, v, z = ϵK∗HGδα t − v, x − z 71

Then, there exists β0 > 0 satisfying

If R ≤ ρ2/2CF C2 V and ρ ≤ K v C v CF , by Lemma2, we can get

P supt∈ 0,T

ϵt

0 ℝK∗HG

δα t − s, x − z W ds, dz

2≥ ρ

≤ 2 T2π2 + 1 exp −Rϵ2

73

The proof of Theorem 3 is completed.

In the following, we first give Garsis-Rodemich-Rumser’sLemma in Bally et al. [29].

Lemma 4. Assume p > 1 and Uε t, x : t, x ∈ 0, T ×Dare a family of real-valued stochastic processes. Suppose thefollowing is true.

Assumption 3. One has

limε→0

E Uε t, x p = 0 74

as t, x ∈ 0, T ×D.

limϵ→0

sup ϵ2 log P supt∈ 0,T

ϵt

0 ℝK∗HG

δα t − s, x − z W ds, dz

2≥ ρ, ϵW ∞ < δ ≤ −R 67

t

0 DK∗H G

δα t − v, x − z −G

δα s − v, y − z

2dvdz

≤ Ct

0 ℝK∗H G

δα t − v, x − z − G

δα s − v, x − z

2dvdz +

t

0 ℝK∗H G

δα s − v, x − z − G

δα s − v, y − z

2dvds

≤ Ct

o ℝGδα t − v, x − z −G

δα s − v, x − z

1/Hdvds

2H+

t

0 ℝGδα s − v, x − z −G

δα s − v, y − z

1/Hdvdz

2H

≤ C t − s μ + x − z θ β0

72

8 Complexity

Assumption 4. For any t, x , s, z ∈ 0, T ×D, there existsν > 0 satisfying

E U ε t, x −U ε s, z p ≤ C t − s + x − z 2 γ 75

Then, for any r ∈ 1, p , γ ∈ 0, β/p ,

limε→0

E Uε γr= 0 76

Now we can prove Theorem 2.

Proof of Theorem 2. Set Uε = vε − v0 / ε. We will prove that

limε→0

E U ε −U γr= 0 77

To this end, we need to prove that Assumptions 3 and 4are satisfied for Wε =U ε −U . Note that

Uε t, x −U t, x

=t

0 DGδα t − s, x − z

⋅f vε s, z − f v0 s, z

ε− f ′ v0 s, z U s, z dsdz

≕ IIIε t, x + IVε t, x78

where

IIIε t, x

=t

0 DGδα t − s, x − z

⋅f vε s, z − f v0 s, z

ε− f ′ v0 s, z U ε s, z dsdz

79

and

IVε t, x =t

0 DGδα t − s, x − z f ′ v0 s, z Uε s, z −U s, z dsdz

80

Using Taylor’s formula, there exists a ξε t, x such that

f vε t, x − f v0 t, x = f ′ v0 t, x + ξε t, x vε t, x − v0 t, x× vε t, x − v0 t, x

81

Note that f ′ is Lipschitz continuous and ξε t, x ∈ 0, 1 ;one can get

f ′ v0 t, x + ξε t, x vε t, x − v0 t, x × vε t, x − v0 t, x≤m′ξε t, x vε t, x − v0 t, x ≤m′ vε t, x − v0 t, x

82

So

IIIε t, x ≤m′t

0 DGδα t − s, x − z v

ε t, x − v0 t, x Uε s, z dsdz

= εm′t

0 DGδα t − s, x − z U

ε s, z 2dsdz

83

Using Hölder’s inequality, for p < 2 and 1/α < q < α, onecan get

E IIIε t∞p ≤ εp/2m′p sup

t,x ∈ 0,T ×D

t

0 DGδα t − s, x − z

qdsdz

p/q

×t

0E Uε s∞

2pds,

84

where 1/p + 1/q = 1. Together with (5) of Lemma A.1 andProposition 1, there exists a constant C p,m,m′, T onlydepending on p,m,m′, T satisfying

E IIIε t∞p ≤ εp/2C p,M,M , T 85

Since f ′ ≤M, for p > 2, together with Hölder’sinequality and (5) of Lemma A.1, we can get

E IVε t∞p ≤Mp sup

t,x ∈ 0,T ×D

t

0 DGδα t − s, x − z

qdsdz

p/q

× Et

0Uε −U s∞

pds,

86

where 1/p + 1/q = 1. Together with (52), (56), and (57),we can get

E U ε −U t∞p ≤ C p,m,m , T εp/2 + E

t

0U ε −U s∞

pds

87

Using Gronwall’s inequality, one can get

E Uε −U t∞p ≤ εp/2C p,m,m , T → 0, ε→ 0, 88

which implies Assumption 3 in Lemma 4.

9Complexity

Now we prove Assumption 4 in Lemma 4. We willprove IIIε and IVε satisfy Assumption 4 in Lemma 4.Using Hölder’s inequality and (A.9), one can get

where θ′ ∈ 0, min 1, α + 1 /q − 1 and 1/p + 1/q = 1.Similarly,

where μ ∈ 0, α + 1/q − 1 /α . Together with (89) and (90),we can get

E IIIε t, x − IIIε s, z p ≤ C t − s + x − z 2 β 91

Similarly, one can get

E IVε t, x − IVε s, z p ≤ C t − s + x − z 2 β 92

Together with (91) and (92), we can get

E Uε t, x −U t, x − Uε s, z −U s, z p

≤ C t − s + x − z 2 β93

Then, for any p > 2 and q ∈ 1, α such that γ ∈ 0, β/pand r ∈ 1, p , using Lemma 4,

limε→0

E V ε γr= 0 94

The proof of Theorem 2 is completed.

Proof of Theorem 3. By Theorem 1, U/a ε obeys largedeviation principles on Cγ 0, T ×D , with the rate functionI given by (30) and the speed function h2 ε . Using Demboand Zeitouni ([30] Theorem 4.2.13), to prove the largedeviation principles of Uε/a ε is e2 ε -exponentially equiva-lent to U/a ε , that is,

lim supε→0

e−2 ε log PU ε −U μ′,θ′

a ε> δ = −∞ 95

holds for any δ > 0.Since

U ε −U γ ≤ C T , µ′, θ′ Uε −U T ,γ 96

To prove (67), we only need to prove

lim supε→0

e−2 ε log PU ε −U t, μ′, θ′

a ε> δ = −∞ 97

Note the decomposition

Uε t, x −U t, x = IIIε t, x + IVε t, x 98

E IIIε t, x − IIIε t, xp≤ E εm

t

0 DGδα t − s, x − z − G

δα t − s, x − z U

ε s, z 2dsdzp

≤ εp/2M pt

0 DGδα t − s, x − z −G

δα t − s, x ‐z

qdsdz

p/q×

t

0 DE Yε s, z s∞

2pdsdz1/p

≤ εp/2C p,m,m , T x − xpθ,

89

E IVε t, x − IVε t , xp≤ E εm

t

0 DGδα t − s, x − z −G

δα t ‐s, x‐z U ε s, z 2dsdz

p

≤ εp/2C p,m, Tt

0 DGδα t − s, x − z −G

δα t ‐s, x‐z U ε s, z 2dsdz

p

≤ εp/2C p,m, Tt

t DGδα t − s, x − z − G

δα t ‐s, x‐z U ε s, z 2dsdz

p

+ εp/2C p,m, Tt

0 DGδα t − s, x − z U

ε s, z2

dsdz p ≤ C p,m,m , T t − tpμ,

90

10 Complexity

For any q ∈ 1, α , x, x′ ∈ 0, 1 , 1/p + 1/q = 1, and0 ≤ s ≤ t ≤ T , by Hölder’s inequality, (32), and (A.9), wecan get

where θ′ ∈ 0, min 1, α + 1 /q − 1 . Similarly,

where μ′ ∈ 0, α + 1 /q − 1/α . Together with (99) and (100),we can get

E IVε t, x − IVε s, z p ≤ C t − s µ′ + x − z θ′ 101

Note that U ε −U s,∞ ≤ C s, µ′, θ′ Uε −U s,γ, we have

IVε t, x t,γ ≤ C C s, μ′, θ′ U ε −U s,γpds 102

Thus, for t ∈ 0, T , we have

Uε −U t,γp≤ C p,m, T IIIε t,γ +

t

0Uε −U s,γ

pds

103

Applying Gronwall’s Lemma to f t = Uε −U t,γp,

we have

U ε −U T ,γp≤ C p, T ,m IIIε T ,γ

p104

By (64) to prove (67), we only need to prove that for anyδ > 0,

lim supε→0

e−2 ε log PIε T,γa ε

≥ δ = −∞ 105

Note that

IIIε t, x =t

0 DGδα t − s, x − z A

ε s, z dsdz, 106

where

Aε s, z =f vε s, z − f v0 s, z

ε− f ′ v0 s, z U ε s, z

107

By the same method in the proof of (59), we have

IIIε T ,γ ≤ C T, µ′, θ′ Aε T ,∞ 108

IVε t, x − IVε t, x′ ≤m′t

0 DGδα t − s, x − z −G

δα t − s, x′ − z Uε s, z −U s, z dsdz

≤m′t

0 DGδα t − s, x − z −G

δα t − s, x′ − z

qdsdz

1/q×

t

0 DUε s, z −U s, z pdsdz

1/p

≤M′ x − x′ θ ×t

0U ε −U u∞

p1/p

99

IVε t, x − IVε t′, x ≤m′t

0 DGδα t − s, x − z − G

δα t ‐s, x‐z U ε s, z −U s, z dsdz

≤m′t

0 DGδα t − s, x − z −G

δα t − s, x − z

qdsdz

1/q×

t

0 DUε s, z −U s, z pdsdz

1/p

+m′t

t DGδα t − s, x − z

q

dsdz1/q

×t

t DU ε s, z −U s, z pdsdz

1/p

≤m′ t − t′ μ′ ×t

0Uε −U u∞

p1/p

100

11Complexity

Similar to the proof of (53), we have

Aε T ,∞ ≤m′vε − v0

T ,∞

2

ε109

Together with (40) and (A.11), for any t ∈ 0, T , we canget that

vε − v0t,∞

≤ supt,x ∈ 0 T ×D

l

0 DGδα l − s, x − z m v

ε s, z − v0 s, z dsdz

supt,x ∈ 0,T ×D

εl

0 DGδα t − s, x − z B

H dsdz

≤ C m, Tt

0vε − v0

s,∞ds + IIVεt,∞,

110

where

IIVε t, x = ε1

0 DGδα t − s, x − z B

H dsdz

= ε1

0 DK∗HG

δα t − s, x − z W dsdz

111

By Gronwall’s Lemma, one can get

vε − v0T ,∞ ≤ C M, T IIV

εT,∞ ≤ C M, T IIV

ε γ 112

Applying Lemma 4 with

F t, x, s, z = K∗HG t − s, x − z 1s≤t , β0 =12, CF = C,

ρ = εM 1 + v0T ,∞ + θ ,N t, x = ε1 vε T ,∞< v0 T,∞+θ ,

113

for any fixed θ > 0, one can get that for any β,G ≥ εK 1 +v0∣T ,∞ + θ C1/2F Ç β, β0 C β, β0 ,

P IIVε∣β,β ≥M, vε T ,∞ < v0

T,∞ + θ

≤ L exp −G2

εM2CC2 β, β0 1 + v0∣T ,∞ + θ2

114

We can get

lim supε→0

e−2 ε log P vε∣T ,∞ ≥ v0 T ,∞ + θ

≤ lim supε→0

e−2 ε log P vε − v0∣T ,∞ ≥ θ = −∞

115

Together with (73), (78), (82), (83), and (85), we have

This completes the proof.

Appendix

Green Function

The nonlocal factional differential operator Dδ,α isdefined by

ℱ Dδ,αϕ ξ = − ξ α exp −iδπ

2sgn ξ ℱ ϕ ξ , A 1

where α is called the index of stability and δ improperlyreferred to as the skewness which satisfy δ ≤min α − α 2,2 + α 2 − α and δ = 0 when δ ∈ 2ℕ + 1.

The operatorDδ,α is a closed, densely defined operator onL2 ℝ , and it is the infinitesimal generator of a semigroupwhich is in general not symmetric and not a contraction. Thisoperator is a generalization of various well-known operators,such as the Laplacian operator (when α = 2), the inverse ofthe generalized Riesz-Feller potential (when α > 2), and the

lim supε→0

e−2 ε log PIII T,μ ,θa ε

> δ

≤ lim supε→0

e−2 ε log P III T,μ ,θ2>

εa ε δC β, T, m

≤ lim supε→0

e−2 ε log P III T, μ , θ2>

εa ε δC β, T, m

, vε T,∞ < v0

T,∞ + θ + P vεT ,∞ ≥ v

0T ,∞ + θ

≤ lim supε→0

−δεa ε C β, T ,m m2CC2 β, β0 1 + v0∣T ,∞ + θ

2 ∨ lim supε→0

e−2 ε log P vε∣T ,∞ ≥ v0 T ,∞ + θ = −∞

116

12 Complexity

Riemann-Liouville differential operator (when δ = 2 + α 2or δ = α − α ). It is self-adjoint only when δ = 0, and in thiscase, it coincides with the fractional power of the Laplacian.We refer the readers to Debbi and Dozzi [31] for more detailsabout this operator.

According to Komatsu [32], Dδ,α can be represented for1 < α < 2, by

Dδ,αφ x

=1

0

φ x + y − φ x − yφ′ xy 1+α

κδ−1 −∞,0 y + κδ+1 −0,+∞ y dy,

A 2

and for 0 < α < 1, by

Dδ,αφ x =1

0

φ x + y − φ xy 1+α

κδ−1 −∞,0 y + κδ+1 −0,+∞ y dy,

A 3

where κδ− are κδ+ two nonnegative constants satisfying κ

δ− +

κδ+ > 0, φ is a smooth function for which the integral exists,and φ′ is its derivative. This representation identifies it asthe infinitesimal generator for a nonsymmetric α-stableLévy process.

Suppose Gδα t, x is the fundamental solution to thefollowing equation:

∂v∂t

t, x =Dδ,α t, x ,

v 0, x = δ0 x , t > 0, x ∈ℝ,A 4

where δ0 · is the Dirac distribution. Using Fourier trans-form, one can get Gδα t, x which is given by

Gδα t, x =12π D

exp −izx − t z α exp −iδπ

2sgn z dz

A 5

Let us list some known facts on Gδα t, x which will be usedlater on (see, e.g., Debbi and Dozzi [31]).

Lemma A.1. Suppose α ∈ 0,∞ / ℕ , one can get thefollowing:

(1) Gδα t, · is not symmetric with respect to x.

(2) For any x ∈ℝ and s, t ∈ 0,∞ ,

∂n

∂xnGδα t, x = s

− n+t /α Gδα s−1t, s− 1/α x A 6

or equivalently

∂n

∂xnGδα t, x = t

− n+t /α Gδα 1, t− 1/α x A 7

(3) For any s, t ∈ 0,∞ , Gδα s, ⋅ ∗Gδα t, ⋅ =Gδα s + t, ⋅

(4) For n ≥ 1, there exist constants C and Cn > 0satisfying

Gδα 1, x ≤ C1

1 + x 1+α, 

∂n

∂xnGδα 1, x ≤ Cn

x α+n−1

1 + x α+n 2

A 8

(5) T0 D Gδα t, x

λdtdx 0 is a constant.

Proof. For any x, y ∈ℝ and t ∈ 0, T ,

t

0 ℝGδα t − v, x −w − G

δα t − v, y −w

1/Hdwdv

H

= Gδα t − ∗, x − ⋅ −Gδα t − ∗, y − ⋅

L1/H 0,T ×ℝ

= Gδα t − ∗, x − ⋅ −Gδα t − ∗, y − ⋅

θ

⋅ Gδα t − ∗, x − ⋅ − Gδα t − ∗, y − ⋅

1−θ

L1/H 0,T ×ℝ

≤ Gδα t − ∗, x − ⋅ −Gδα t − ∗, y − ⋅

θ

⋅ Gδα t − ∗, x − ⋅1−θ

L1/H 0,T ×ℝ

+ Gδα t − ∗, x − ⋅ −Gδα t − ∗, y − ⋅

θ

⋅ Gδα t − ∗, y − ⋅1−θ

L1/H 0,T ×ℝ≔ I1 + I2

A 12

13Complexity

By the mean-value theorem, for θ ∈ 0, 1 , one canget that

and

ℝ

∂∂x

Gδα t − u, η −wθ/H

Gδα t − u, x −w1−θ /H

dw

= t − u 1/α − 1+θ /αHℝ

∂∂x

Gδα 1, η −wθ/H

Gδα 1,w1−θ /H

dw

≤ CH,θ t − u1/α − 1+θ /αH

A 14

Therefore, if 1/α − 1 + θ /αH + 1 > 0, that is, θ <α + 1 H − 1, then,

I1 ≤ CT ,θ,H x −wθ A 15

Similarly, we can check

I2 ≤ CT ,θ,H x −wθ A 16

Hence, the inequality (A.9) holds. As for the inequality(A.10), for any x ∈ℝ and t, s ∈ 0, T ,

t

0 ℝGδα t − v, x −w − G

δα s − v, x −w

1/Hdzdv

H

= Gδα t − ∗, x − ⋅ −Gδα s − ∗, x − ⋅

L1/H 0,T ×ℝ

= Gδα t − ∗, x − ⋅ − Gδα s − ∗, x − ⋅

μ

⋅ Gδα t − ∗, x − ⋅ −Gδα s − ∗, x − ⋅

1−μ

L1/H 0,T ×ℝ

≤ Gδα t − ∗, x − ⋅ −Gδα s − ∗, x − ⋅

μ

⋅ Gδα s − ∗, x − ⋅1−μ

L1/H 0,T ×ℝ

+ Gδα t − ∗, x − ⋅ −Gδα s − ∗, x − ⋅

μ

⋅ Gδα s − ∗, x − ⋅1−μ

L1/H 0,T ×ℝ≔ II1 + II2

A 17

By mean-value theorem, it holds that

Note that

Gδα t, y = t−1/αGδα 1, t

− 1/α y A 19

Hence, one can attain

∂∂t

Gδα t, y = −1α

t− 1− 1/α Gδα 1, t− 1/α y

+ −1α

t− 1−2 1/α y∂∂z

Gδα 1,w w=t− 1/α y

A 20

So

I1 = x − y θ∂∂x

Gδα t − ∗, η − ⋅θ

⋅ Gδα t − ∗, x − ⋅1−θ

L1/H 0,T ×ℝ

= x − y θ/H∂

0 ℝ

∂∂x

Gδα t − u, η −wθ/H

Gδα t − v, x −w1−θ /H

dwdu

HA 13

II1 = t − s μ∂∂t

Gδα ρ − ∗, x − ⋅μ

⋅ Gδα t − ∗, x − ⋅1−μ

∥L1/H 0,T ×ℝ

=T

0 ℝt − s ∣ μ/H

∂∂t

Gδα ρ − r, x −wμ/H

Gδα t − r, x −w1−μ /H

dwdu

H A 18

14 Complexity

Therefore, if − αμ + 1 /Hα + 1/α + 1 > 0 i.e. μ <α + 1 H − 1 /α, then, for μ ∈ 0, a + 1 H − 1 /α ,

II1 ≤ Cα,H,T,µ t − s µ A 22

Similarly,

II2 ≤ Cα,H,T,µ t − s µ A 23

So (A.10) holds. Now let us prove that (A.11) holds. Forany x ∈ℝ and s, t ∈ 0, T ,

t

s ℝGδα t − v, x −w

1/Hdwdu

H

≤t

st − u − 1/αH + 1/α

ℝGδα 1, z

1/Hdw dv

H

≤ CH t − sa+1 H−1 /α

A 24

The proof of the lemma is completed.

Lemma A.3. Suppose p ∈ 1,∞ , q ∈ 1, p , and ρ ∈ 1,∞satisfying

1ρ=1p−1q+ 1 ∈ 0, 1 A 25

Suppose Gδα = Gδα t, x − z is the Green kernel, Q = ∂/∂zGδα orQ = Gδα with t, x, z ∈ 0, T ×ℝ ×ℝ. We define I by

I u t, x =t

0 ℝQ t − s, x − z u s, z dsds, A 26

with u ∈ L1 0, T ; Lq . Then, I L1 0, T ; Lq →L∞ 0, T ; Lq is a bounded linear operator which satisfiesthe following:

(1) When Q = ∂/∂y Gδα, then, for all r ∈ 1, 1 + α/2 ,

I u t, ⋅ p

≤ Ct

0t − s − 2/α + 1/αr u x, ⋅ qds, ∀t ∈ 0, T

A 27

(2) When Q = Gδα, then, for all ρ ∈ 1, 1 + α ,

I u t, ⋅ p

≤ Ct

0t − s − 1/α + 1/αr u x, ⋅ qds, ∀t ∈ 0, T

A 28

Proof. Together withMinkowski’s inequality, Young inequal-ity and (4) of Lemma A.1. We can get

I u t, ⋅ p =t

0 ℝ

∂∂z

Gδα t − s, ⋅ − z v s, z dzdsp

≤t

0 ℝ

∂∂z

Gδα t − s, ⋅ − z u s, z dzp

ds

≤ Ct

0t − s − 2/a

ℝ

∂∂z

Gδα 1, t − s− 1/a ⋅ −z u s, z dz

p

ds

≤ Ct

0t − s − 2/a

∂∂z

Gδα 1, t − s− 1/a ⋅ ∗ u s, ⋅ ⋅

p

ds

≤ Ct

0t − s − 2/a

∂∂y

Gδα 1, t − s− 1/a ⋅

p

⋅ u s, ⋅ qds

≤ Ct

0t − s − 2/a + 1/αρ u s, ⋅ qds,

A 29

ℝ

∂∂t

Gδα ρ − r, x −wμ/H

Gδα t − r, x −w1−μ /H

dw

≤ Cα,Hℝt − r −1− 1/α μ/H − 1/α 1−u /H Gδα 1, ρ − u

− 1/α x −wμ/H

⋅ Gδα 1, t − u− 1/α x −w

1−μ /Hdw

+ Cα,Hℝt − r −1− 2/α μ/H − 1/α 1−u /H x −w μ/H

∂∂w

Gδα 1,w ∣w= ρ−r − 1/α x−wμ/H

⋅ Gδα 1, t − u− 1/α x −w

1−μ /Hdw

≤ Cα,H t − u− αμ+1 /Hα + 1/α + Cα,H t − u

− αμ+μ+1 /Hα + 1/α

A 21

15Complexity

where one has used the result that, for ρ ∈ 1, 1 + α /2 ,

Gδα 1, t − s− 1/α

ρ=

ℝ

∂∂z

Gδα 1, t − s− 1/α z

ρdz

1/p

≤ t − s 1/αρℝ

∂∂y

Gδα 1, yρdy

1/p

≤ C t − s 1/αρ

A 30

The proof of (2) is omitted since it is similar to case (1).The proof of this lemma is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this article.

Acknowledgments

Xichao Sun is partially supported by the Natural ScienceFoundation of Anhui Province (1808085MA02) and the Nat-ural Science Foundation of Bengbu University (2017GJPY04,2016KYTD02, 2017jxtd2, and 2017ZR08). Ming Li is par-tially supported by National Natural Science Foundation ofChina under the project Grant nos. 61672238, 61272402,and 61070214.

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