# Moderate Deviations for Stochastic Fractional Heat Equation Driven by Fractional Noise

Authors: Xichao Sun; Ming Li; Wei Zhao
Journal: Complexity
 (2018)

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## Abstract

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


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## Body

## 1. Introduction

Since the work of Freidlin and Wentzell [1], the large deviation principle (LDP) has been extensively developed for small noise systems and other types of models (such as interacting particle systems) (see [2–7]). Cardon-Weber [2] proved a LDP for a Burgers-type SPDE driven by white noise. Marquez-Carreras and Sarra [3] proved a LDP for a stochastic heat equation with spatially correlated noise, and Mellali and Mellouk [4] extended Marquez-Carreras and Sarra’s [3] to a fractional operator. Jiang et al. [5] proved a LDP for a fourth-order stochastic heat equation driven by fractional noise. Budhiraja et al. [6] studied large deviation properties of systems of weakly interacting particles. Budhiraja et al. [7] proved a large deviation for Brownian particle systems with killing.Similar to the large deviation, the moderate deviation problems also come from the theory of statistical inference. Using the moderate deviation principle (MDP), we can get the rate of convergence and an important method to construct asymptotic confidence intervals, for example, Liming [8], Guillin and Liptser [9], Cattani and Ciancio [10], and other references therein. There are also many works about MDP about stochastic (partial) differential equations; some surveys and literatures could be found in 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 many physicists and mathematicians due to various applications in risk management, image analysis, and statistical mechanics (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 applications). Stochastic partial differential equations involving a fractional Laplacian operator have been studied by many authors; see Mueller [22], Wu [23], Liu et al. [24], Wu [25], and the references therein.Motived above, we investigated the moderate deviations about the stochastic fractional heat equation with fractional noise as follows:(1)∂vε∂tt,x=Dδ,αvεt,x+fvεt,x+εBHdt,dx,vεo,x=0,where t∈0,T, x∈D=0,1, Dδ,α is the fractional Laplacian operator which is defined in Appendix, and BHdt,dx denotes a fractional noise which is fractional in time and white in space with Hurst parameter H∈1/2,1; that is, BH is a mean zero Gaussian random field on 0,T×D with covariance.
(2)CovBHt,xBHs,y=12t2H+s2H−t−s2Hx∧y.Assume that the coefficients satisfy the following.Assumption 1.
Functionf is Lipschitz; that is, there exist an m>0 satisfying
(3)fy−fx≤my−x,∀x,y∈D.Under the conditions of Assumption1, (1) possesses a unique solution in the sense of Walsh [26] as follows:
(4)vεt,x=∫0t∫DGαδt−s,x−zfvεs,zdzds+ε∫0t∫DGαδt−s,x−zBHds,dz.As the parameterε→0, the solution vεt,x of (1) will tend to v0t,x which is the solution to the following equation:
(5)∂v0∂tt,x=Dδ,αv0t,x+fv0t,x,v00,x=0.This paper mainly devotes to investigate the deviations ofvε from the deterministic solution v0, as ε→0, that is, the asymptotic behavior of the trajectories.
(6)Vεt,x≔1εaεvε−v0t,x,t,x∈0,T×D,where aε is the same deviation scale that strongly influences the asymptotic behavior of Vε.Ifaε=1/ε, we are in the domain of large deviation estimate, which can be proved similarly to Jiang et al. [5].The caseaε≡1 provides the central limit theorem. As ε↓0, we will prove that vε−v0/ε converges to a random field in this paper.To fill the gap between scaleaε=1 and scale aε=1/ε, we mainly devote to the moderate deviation when the scale satisfies the following:
(7)aε→+∞,εaε→0,ε→0.This paper is organized as follows. In Section2, the definition of the fractional noise BHds,dz is given. In Section 3, the main result is given and proved. In Appendix, some results about the Green kernel are given.


## 2. Fractional Noise

LetH∈1/2,1, and BH0,t×At,A∈0,T×Bℝ is a centered Gaussian family of random variables with the covariance satisfying
(8)EBHt,ABHs,B=A∩BRHt,s,with s,t∈0,T,A,B∈Bℝ and covariance kernel RHt,s=1/2t2H+s2H−t−s2H, where ∣A∣ denotes the Lebesgue measure of the set A∈Bℝ and Bℝ denotes the class of Borel sets in ℝ.We denoteφ as the set of step functions on 0,T×ℝ. Let H be the Hilbert space defined as the closure of φ with respect to the scalar product.
(9)10,t×A,10,s×BH=EBHt,ABHs,B.According to Nualart and Ouknine [27], the mapping 10,t×A→BHt,A can be extended to an isometry between H and the Gaussian space H1BH associated with BH and denoted by
(10)ϕ↦BHϕ≔∫0t∫Aϕs,xBHds,dx.Define the linear operatorKH∗:φ↦L20,T by
(11)KH∗ϕ=KHT,sϕs,y+∫sTϕr,y−ϕs,y∂KH∂rr,sdr,where KH is defined by
(12)KHt,s≔CHs1/2−H∫stu−sH−3/21−su1/2−Hdu+CHt−sH−1/2,0<s≤t,0,otherwise,with CH=2αΓ3/2−α/Γα+1/2Γ2−2α1/2, and one can get
(13)∂KH∂tt,s=CH12−Ht−sH−3/2tsH−1/2.Moreover,KH satisfies the following:
(14)∫0s∧tKHs,rKHt,rdr=12t2H+s2H−t−s2H.Then, since(15)KH∗10,t×As,x=KHt,s10,t×As,x,one can get
(16)KH∗ϕ,KH∗ψL20,T×D=ϕ,ψℋ,where ϕ and ψ in φ are any step functions. So the operator KH∗ gives an isometry between the Hilbert space ℋ and L20,T×0,1. Hence, Wt,A,t∈0,T,A∈ℬ0,1 defined by
(17)Wt,A=BHKH∗−110,t×Ais a space-time white noise, and BH has the following form:
(18)BHt,x=∫0t∫0xKHt,sWds,dy.Therefore, the mild formulation of (4) has the following form:
(19)vεt,x=∫0t∫DGαδt−s,x−zfvεs,zdzds+ε∫0t∫DKH∗Gαδt−s,x−zWds,dz.That is, the last term of (4) is equal to
(20)∫0t∫DGαδt−s,x−zBHds,dz=∫0t∫DKH∗Gαδt−s,x−zWds,dz.The following embedding proposition is given by Nualart and Ouknine [27].Lemma 1.
SetH>1/2, then, we have
(21)L1/H0,T×ℝ⊂ℋ.


## 3. Main Results and Their Proof

### 3.1. Main Results

For any functionϕ defined on 0,T×D, let
(22)ϕt,∞=supϕs,z:s,z∈0,t×D,ϕt,γ=supϕs1,z1−ϕs2,z2s1−s2μ+z1−z2θγ:s1,z1≠s2,z2∈0,t×D,where μ∈0,α+1H−1/α, θ∈0,min1,α+1H−1, and γ∈0,1. Let
(23)ϕγ=ϕT,∞+ϕT,γ.LetCγ0,T×D be the functions ϕ:0,T×D which satisfy ϕγ<∞, endowed with the ϕγ−norm.Define(24)E=ex,t=∫0t∫0xe¯s,zdsdz,e¯∈L20,T×D,which is a Cameron-Martin space endowed with the norm
(25)eℰ=∫0t∫0xe¯2s,zdzds1/2.Supposee∈E. Now, let ve be the solution of the following deterministic equation:
(26)∂ve∂tt,x=Dδ,αvet,x+fvet,x+Fe¯t,x,ve0,x=0,where
(27)Fe¯x,t≔∂2∂x∂t∫0t∫0xKHt,se¯z,sdzdssatisfies the following:
(28)∫0t∫DGαδt−s,x−zFe¯z,edzds=∫0t∫DKH∗Gαδt−s,x−ze¯z,sdzds.The above (26) shows a unique mild solution.
(29)Zex,t=∫0t∫DGαδt−s,x−zfs,z,Zez,sdzds+∫0t∫DKH∗Gαδt−s,x−ze¯z,sdzds.Similar to Jiang et al. [5], one can get the following:Theorem 1.
LetH∈1/2,1. Under Assumption1, the law of the solution to (1) satisfies a deviation principle onCγD×0,Twith the good rate function:
(30)Iφ=inf12eℰ2;Ze=φ,with the conventioninf∅=∞, where0<γ<1.
More precisely, for any Borel measurable subset B ofCγ0,T×D,
(31)−infφ∈BoIφ≤liminfε→0εlogPZe∈B≤limsupε→0εlogPZe∈B≤−infφ∈B¯Iφ,whereBoandB¯denote the interior and the closure ofB, respectively.We furthermore suppose that the coefficients satisfy the following.Assumption 2.
f is differentiable, and the derivative f′ of f is Lipschitz. That is to say, there exist positive constant m and m′ which satisfy the following:
(32)f′y−f′x≤m′y−x,∀x,y∈D.Together with the Lipschitz off, we conclude that
(33)f′x≤m.Now, we give the following central limit theorem.Theorem 2.
Letfand its derivativef′satisfy Assumptions 1and2. Then, forp≥1,ve−v0/εconverges inLpto a random fieldUonCγ0,T×Dwith0<γ<1, determined by
(34)∂Ut,x∂t=Dδ,αUt,x+f′v0t,xUt,x+BHdt,dx,U0,x=0,for all,t,x∈0,T×D.Let the functionUe be the solution to the following partial differential equation:
(35)∂Uet,x∂t=Dδ,αUet,x+f′v0t,xUet,x+Fe¯t,x,Ue0,x=0.Under Assumptions1 and 2, by Theorem 1, one can get U/aε which satisfies large deviation principles on Cγ0,T×D with the speed e2ε and the good rate function satisfies the following:
(36)Iφinf12eℰ2,Ue=φ,+∞,otherwise.Now, the second result is given as follows:Theorem 3.
In moderate deviation principle, letH∈1/2,1. Under the Assumptions1and2, then, the random field1/εaεvε−v0satisfies a large deviation principle on the spaceCγ0,T×Dwith speeda2εand the good rate functionIφdefined by (36), where0<γ<1.


### 3.2. Convergence of the Solution

Lemma 2.
LetH∈1/2,1. Under Assumption1, then, there exists a unique solution to (1). Moreover, for anyp∈1,∞,T>0,
(37)supε≥1supt,x∈0,T×DEvεt,xp<∞.Asε→0, we get the convergence of vε as follows:Proposition 1.
LetH∈1/2,1andp≥2. By Assumption1, there exists a constantcm,p,H,T which satisfies the following:
(38)Evε−v0t,∞p≤εp/2cm,p,H,T.Proof.
Note that(39)vεt,x−v0t,x=∫0t∫DGαδt−s,x−zfvεs,z−fv0s,zdzds+ε∫0t∫DGαδt−s,x−zBHdsdz.One can get
(40)vεt,x−v0t,xT,∞p≤2p−1supt,x∈0,T×D∫0t∫DGαδt−s,x−zfvεs,z−fv0s,zdzdsp+εp/22p−1supt,x∈0,T×D∫0t∫DGαδt−s,x−zBHdsdzp.Set
(41)N1εt,x=∫0t∫DGαδt−s,x−zfvεs,z−fv0s,zdzds,N2εt,x=∫0t∫DGαδt−s,x−zBHdsdz.
Together with Hölder’s inequality, the Lipschitz condition (C) and (5) of LemmaA.1, for 1/α<q<α, we have
(42)EN1εT,∞p≤mpsupt,x∈0,T×DGαδt−s,x−zqdzdsp/q×E∫0Tvε−v0T,∞pdt≤Cp,m,TE∫0Tvε−v0T,∞pdt,where 1/p+1/q=1.
For any0≤t≤T, x,z∈D, p>2, by (21) and (A.9), there exist θ∈0,min1,α+1H−1 which satisfies the following:
(43)EN2εt,x−N2εt,x′p=E∫0t∫DGαδt−s,x−z−Gαδt−s,x′−zBHdsdzp≤∫0t∫DKH∗Gαδt−s,x−z−Gαδt−s,x′−z2dsdzp/2≤∫0t∫DGαδt−s,x−z−Gαδt−s,x′−z1/HdsdzHp≤mx−x′pθ.
Similarly,(44)EN2εt,x−N2εt′,xp=E∫0t∫DGαδt−s,x−z−Gαδt′−s,x−zBHdsdzp≤∫0t∫DKH∗Gαδt−s,x−z−Gαδt′−s,x−z2dsdzp/2≤Cp,m,T∫t't∫DKH∗Gαδt−s,x−z2dsdzp/2+Cp,m,T∫0t'∫DKH∗Gαδt−s,x−z−Gαδt'−s,x−z2dsdzp/2.
By (21), (A.10), and (A.11), for μ∈0,α+1H−1/α, one can get
(45)∫t't∫DKH∗Gαδt−s,x−z2dsdzp/2≤CH∫t't∫DGαδt−s,x−z1/HdsdzHp≤CHt−t′pα+1H−1/αand
(46)∫0t'∫DKH∗Gαδt−s,x−z−Gαδt′−s,x−z2dsdzp/2≤CH∫t't∫DKH∗Gαδt−s,x−z−Gαδt′−s,x−z1/HdsdzHp≤CHt−t′pμ.
Together with (43), (44), (45), and (46), one can get for any p≥2 and t′,x′,t,x∈0,T×D, there exists a constant β>0 such that
(47)EN2εt,x−N2εt,x′p≤Cp,m,H,Tx−x′θ+t−t′μpβ,where Cp,m,H,T is independent of ε. For p>2/β, by Garsis-Rodemich-Rumser’s Lemma, there exist a constant C and a random variable Mp,εω satisfying
(48)N2εt,x−N2εt,x′p≤mp,εωx−x′θ+t−t′μpβ−2logcx−x'θ+t−t'μ2and
(49)supε∈0,1Emp,ε<∞.
If we chooset′=0 in (48), we can get
(50)Esupt,x∈0,T×D∫0t∫DGαδt−s,x−zBHdsdzp≤cm,p,H,Tsupε∈0,1Emp,ε<∞.
By Gronwall’s inequality and (40), (42), and (48), there exists a constant Cp,m,H,T satisfying
(51)Evε−v0t,∞p≤εp/2cm,p,H,T.
The proof is completed.


### 3.3. The Proof of the Main Results

We first prove Theorem1. Our proof is based on the following proposition (see Doss and Priouret [28]).Proposition 2.
SupposeXjϵ,ϵ>0 and j=1,2are two families of random variables andEj,dj and j=1,2are two Polish spaces. Suppose that
(1)
There exists a mapK:I1<∞→E1such thatI1≤a→E1is continuous for anya<∞.
(2)
X1ϵ,ϵ>0satisfies with the rate functionsI1:E1→0,∞.
(3)
For anyR,δ,a>0, there existρ>0andϵ0>0satisfyingI1≤aandϵ≤ϵ0for allh∈E1, and
(52)Pd1X1ϵ≤ρ,d2x2ϵ,Ke≥δ≤exp−Rϵ2.
Then,X2ϵ,ϵ>0satisfies a LDP with the rate functions
(53)Ih=infI1e:Ke=h.To prove Theorem1, one only needs to prove
(i)
Under some topology,Ze:e:eE≤a→C0,T,LpD is continuous for any a>0.
(ii)
In Freidin-Wentzell inequality, for anyR>0, η>0, and e∈E, there exist a δ>0 satisfying
(54)limϵ→0supϵ2logPϵW−e∞<δ,supt∈0,Tuϵt,·−Zet,·∥2≥ρ≤−R.Theorem 4.
When the level seteE≤ais endowed with the topology of uniform convergence on0,T×D,
(55)Ze:e:eE≤a→C0,T,LpDis the continuous map for anya∈0,+∞.Proof.
One only needs to prove that for fixeda>0, e,h∈∥e∥E≤a,
(56)supt∈0,T∫ℝZet,x−Zht,x2dx≤Ce−hℰ.
Note that(57)Zet,x−Zht,x=∫0t∫ℝGαδt−s,x−zfs,z,Zes,z−fs,z,Zhs,zdzds+∫0t∫ℝKH∗Gαδt−s,x−ze¯s,zdzds≔It,x+IIt,x.
Using (A.5) in Appendix with p=q=2, then ρ=1, one can get
(58)It,x22≤C∫0tt−s−1/α+1/αρfs,z,Zes,z−fs,z,Zhs,z2ds≤Csup0≤s≤tZes,⋅−Zhs,⋅2.
Now, we deal withIIt,x, together with
(59)IIt,x22=∫ℝ∫0t∫ℝKH∗Gαδt−s,x−ze¯s,z−h¯s,zdzds2dx≤C∫ℝ∫0t∫ℝKH∗Gαδt−s,x−z2dzdsdx×e−hE2≔CMt×e−hℰ2.
RecallingKH∗ is defined by (11), one can get
(60)KH∗ϕs,x≤maxw∈s,Tϕw,xKHT,s,Mt=∫ℝ∫0t∫ℝKH∗Gαδt−s,x−z2dzdsdx≤∫ℝ∫0t∫ℝmaxw∈s,tGαδt−s,x−z2KH2t,sdzdxds≤C∫0tt−s−1/αt−s2H−1ds<∞.
Using Gronwall’s inequality, we can get(61)supt∈0,T∫ℝZet,x−Zht,x2dx≤Ce−hℰ.
The proof of the theorem is completed.We now prove the Freidin-Wentizell inequality as follows:SupposeWt,x,t,x∈0,T×ℝ is a Brownian sheet. For e∈E and ϵ>0, we define
(62)W~t,x=Wt,x−ex,tϵand
(63)d~pdp=exp1ϵ∫0T∫ℝe¯t,xWdt,dx−12ϵ2∫0T∫ℝe¯2t,xdxdt≔ZT.Using Girsanov’s theorem, the processW~ is a Brownian sheet under P~. Suppose v~εt,x is a solution of (1) under p~. Then,
(64)v~ϵt,x=∫ℝGαδt,x−zv0zdz+∫0t∫ℝGαδt−s,x−zft,x,v~εs,zdzds+∫0t∫ℝKH∗Gαδt−s,x−ze¯s,zdzds+ϵ∫0t∫ℝKH∗Gαδt−s,x−zW~ds,dz.Now, one can prove (36). Note that, under p~, then,
(65)u~ϵt,·−Zet,·2=∫0t∫ℝGαδt−s,·−zft,x,u~εs,z−ft,x,Zes,zdzds2+ϵ∫0t∫ℝKH∗Gαδt−s,·−yW~ds,dz2≤∫0t∫ℝu~εs,·−Zes,·2dzds+ϵ∫0t∫ℝKH∗Gαδt−s,·−zW~ds,dz2.So underp~, by Gronwall’s Lemma, one can get
(66)supt∈0,Tu~εs,·−Zhs,·2≤Csupt∈0,Tϵ∫0t∫ℝKH∗Gαδt−s,x−zW~ds,dz2.Now, one can change (34) the proof to the following theorem.Theorem 5.
Supposee∈Eande∈E≤a. For eachR>0,η>0, ande∈E, there exists a constantδ>0satisfying
(67)limϵ→0supϵ2logP~supt∈0,Tϵ∫0t∫ℝKH∗Gαδt−s,x−zW~ds,dz2≥ρ,ϵW~∞<δ≤−R.In the following, we give a key Lemma to prove Theorem1, which is similar to Candon-Weber [2], and the proof is omitted.Lemma 3.
SupposeF:0,T×D2→ℝ, CF>0andβ0>0satisfying
(68)∫0T∫DFt,x,v,z−Fs,y,v,z2dudz≤CFt−sμ+x−yθβ0,for any s,z and t,x∈0,T×D. Suppose N:Ω×0,T×0,1→ℝ which is an almost surely continuous,Ft-adapted process satisfyingsupNt,x:t,x∈0,T×D≤ρ,a,s, and fort,x∈0,T×D, suppose
(69)Ft,x=∫0T∫DFt,x,v,zNv,zWdvdz.
Then, for any0<β<β0/2, there exist a positive constant C, Ç(β,β0), andCβ,β0such that for allM≥ρCF1/2Ç(β,β0)Cβ,β0,
(70)PFt,β≥M≤Cexp−M2ρ2CzÇβ,β0C2β,β0.Proof of Theorem 5.
Suppose(71)Ft,x,v,z=ϵKH∗Gαδt−v,x−z.Then, there exists β0>0 satisfying
(72)∫0t∫DKH∗Gαδt−v,x−z−Gαδs−v,y−z2dvdz≤C∫0t∫ℝKH∗Gαδt−v,x−z−Gαδs−v,x−z2dvdz+∫0t∫ℝKH∗Gαδs−v,x−z−Gαδs−v,y−z2dvds≤C∫ot∫ℝGαδt−v,x−z−Gαδs−v,x−z1/Hdvds2H+∫0t∫ℝGαδs−v,x−z−Gαδs−v,y−z1/Hdvdz2H≤Ct−sμ+x−zθβ0.
IfR≤ρ2/2CFC2V and ρ≤KvCvCF, by Lemma 2, we can get
(73)P~supt∈0,Tϵ∫0t∫ℝKH∗Gαδt−s,x−zW~ds,dz2≥ρ≤2T2π2+1exp−Rϵ2.
The proof of Theorem3 is completed.In the following, we first give Garsis-Rodemich-Rumser’s Lemma in Bally et al. [29].Lemma 4.
Assumep>1andUεt,x:t,x∈0,T×Dare a family of real-valued stochastic processes. Suppose the following is true.Assumption 3.
One has(74)limε→0EUεt,xp=0as t,x∈0,T×D.Assumption 4.
For anyt,x,s,z∈0,T×D, there existsν>0satisfying
(75)EUεt,x−Uεs,zp≤Ct−s+x−z2γ.
Then, for anyr∈1,p,γ∈0,β/p,
(76)limε→0EUεγr=0.Now we can prove Theorem 2.Proof of Theorem 2.
SetUε=vε−v0/ε. We will prove that
(77)limε→0EUε−Uγr=0.
To this end, we need to prove that Assumptions3 and 4 are satisfied for Wε=Uε−U. Note that
(78)Uεt,x−Ut,x=∫0t∫DGαδt−s,x−zfvεs,z−fv0s,zε−f′v0s,zUs,zdsdz≕IIIεt,x+IVεt,xwhere
(79)IIIεt,x=∫0t∫DGαδt−s,x−zfvεs,z−fv0s,zε−f′v0s,zUεs,zdsdzand
(80)IVεt,x=∫0t∫DGαδt−s,x−zf′v0s,zUεs,z−Us,zdsdz.
Using Taylor’s formula, there exists aξεt,x such that
(81)fvεt,x−fv0t,x=f′v0t,x+ξεt,xvεt,x−v0t,x×vεt,x−v0t,x.
Note thatf′ is Lipschitz continuous and ξεt,x∈0,1; one can get
(82)f′v0t,x+ξεt,xvεt,x−v0t,x×vεt,x−v0t,x≤m′ξεt,xvεt,x−v0t,x≤m′vεt,x−v0t,x.
So(83)IIIεt,x≤m′∫0t∫DGαδt−s,x−zvεt,x−v0t,xUεs,zdsdz=εm′∫0t∫DGαδt−s,x−zUεs,z2dsdz.
Using Hölder’s inequality, forp<2 and 1/α<q<α, one can get
(84)EIIIε∞tp≤εp/2m′psupt,x∈0,T×D∫0t∫DGαδt−s,x−zqdsdzp/q×∫0tEUε∞s2pds,where 1/p+1/q=1. Together with (5) of Lemma A.1 and Proposition 1, there exists a constant Cp,m,m′,T only depending on p,m,m′,T satisfying
(85)EIIIε∞tp≤εp/2Cp,M,M',T.
Sincef′≤M, for p>2, together with Hölder’s inequality and (5) of Lemma A.1, we can get
(86)EIVε∞tp≤Mpsupt,x∈0,T×D∫0t∫DGαδt−s,x−zqdsdzp/q×E∫0tUε−U∞spds,where 1/p+1/q=1. Together with (52), (56), and (57), we can get
(87)EUε−U∞tp≤Cp,m,m',Tεp/2+E∫0tUε−U∞spds.
Using Gronwall’s inequality, one can get(88)EUε−U∞tp≤εp/2Cp,m,m',T→0,ε→0,which implies Assumption 3 in Lemma 4.
Now we prove Assumption4 in Lemma 4. We will prove IIIε and IVε satisfy Assumption 4 in Lemma 4. Using Hölder’s inequality and (A.9), one can get
(89)EIIIεt,x−IIIεt,x'p≤Eεm'∫0t∫DGαδt−s,x−z−Gαδt−s,x'−zUεs,z2dsdzp≤εp/2M'p∫0t∫DGαδt−s,x−z−Gαδt−s,x'‐zqdsdzp/q×∫0t∫DEYεs,z∞s2pdsdz1/p≤εp/2Cp,m,m',Tx−x'pθ',where θ′∈0,min1,α+1/q−1 and 1/p+1/q=1. Similarly,
(90)EIVεt,x−IVεt',xp≤Eεm'∫0t∫DGαδt−s,x−z−Gαδt'‐s,x‐zUεs,z2dsdzp≤εp/2Cp,m,T∫0t∫DGαδt−s,x−z−Gαδt'‐s,x‐zUεs,z2dsdzp≤εp/2Cp,m,T∫t't∫DGαδt−s,x−z−Gαδt'‐s,x‐zUεs,z2dsdzp+εp/2Cp,m,T∫0t'∫DGαδt−s,x−zUεs,z2dsdzp≤Cp,m,m',Tt−t'pμ,where μ'∈0,α+1/q−1/α. Together with (89) and (90), we can get
(91)EIIIεt,x−IIIεs,zp≤Ct−s+x−z2β
Similarly, one can get(92)EIVεt,x−IVεs,zp≤Ct−s+x−z2β.
Together with (91) and (92), we can get
(93)EUεt,x−Ut,x−Uεs,z−Us,zp≤Ct−s+x−z2β.
Then, for anyp>2 and q∈1,α such that γ∈0,β/p and r∈1,p, using Lemma 4,
(94)limε→0EVεγr=0.
The proof of Theorem2 is completed.Proof of Theorem 3.
By Theorem1, U/aε obeys large deviation principles on Cγ0,T×D, with the rate function I given by (30) and the speed function h2ε. Using Dembo and Zeitouni ([30] Theorem 4.2.13), to prove the large deviation principles of Uε/aε is e2ε-exponentially equivalent to U/aε, that is,
(95)limsupε→0e−2εlogPUε−Uμ′,θ′aε>δ=−∞holds for any δ>0.
Since(96)Uε−Uγ≤CT,µ′,θ′Uε−UT,γ.
To prove (67), we only need to prove
(97)limsupε→0e−2εlogPUε−Ut,μ′,θ′aε>δ=−∞.
Note the decomposition(98)Uεt,x−Ut,x=IIIεt,x+IVεt,x.
For anyq∈1,α,x,x′∈0,1, 1/p+1/q=1, and 0≤s≤t≤T, by Hölder’s inequality, (32), and (A.9), we can get
(99)IVεt,x−IVεt,x′≤m′∫0t∫DGαδt−s,x−z−Gαδt−s,x′−zUεs,z−Us,zdsdz≤m′∫0t∫DGαδt−s,x−z−Gαδt−s,x′−zqdsdz1/q×∫0t∫DUεs,z−Us,zpdsdz1/p≤M′x−x′θ×∫0tUε−U∞up1/p.where θ′∈0,min1,α+1/q−1. Similarly,
(100)IVεt,x−IVεt′,x≤m′∫0t'∫DGαδt−s,x−z−Gαδt'‐s,x‐zUεs,z−Us,zdsdz≤m′∫0t∫DGαδt−s,x−z−Gαδt'−s,x−zqdsdz1/q×∫0t'∫DUεs,z−Us,zpdsdz1/p+m′∫t't∫DGαδt−s,x−zqdsdz1/q×∫t't∫DUεs,z−Us,zpdsdz1/p≤m′t−t′μ′×∫0tUε−U∞up1/p.where μ′∈0,α+1/q−1/α. Together with (99) and (100), we can get
(101)EIVεt,x−IVεs,zp≤Ct−sµ′+x−zθ′.
Note thatUε−Us,∞≤Cs,µ′,θ′Uε−Us,γ, we have
(102)IVεt,xt,γ≤C∫Cs,μ′,θ′Uε−Us,γpds.
Thus, fort∈0,T, we have
(103)Uε−Ut,γp≤Cp,m,TIIIεt,γ+∫0tUε−Us,γpds.
Applying Gronwall’s Lemma toft=Uε−Ut,γp, we have
(104)Uε−UT,γp≤Cp,T,mIIIεT,γp.
By (64) to prove (67), we only need to prove that for any δ>0,
(105)limsupε→0e−2εlogPIεT,γaε≥δ=−∞.
Note that(106)IIIεt,x=∫0t∫DGαδt−s,x−zAεs,zdsdz,where
(107)Aεs,z=fvεs,z−fv0s,zε−f′v0s,zUεs,z.
By the same method in the proof of (59), we have
(108)IIIεT,γ≤CT,µ′,θ′AεT,∞.
Similar to the proof of (53), we have
(109)AεT,∞≤m′vε−v0T,∞2ε.
Together with (40) and (A.11), for any t∈0,T, we can get that
(110)vε−v0t,∞≤supt,x∈0.T×D∫0l∫DGαδl−s,x−zmvεs,z−v0s,zdsdzsupt,x∈0,T×Dε∫0l∫DGαδt−s,x−zBHdsdz≤Cm,T∫0tvε−v0s,∞ds+IIVεt,∞,where
(111)IIVεt,x=ε∫01∫DGαδt−s,x−zBHdsdz=ε∫01∫DKH∗Gαδt−s,x−zWdsdz.
By Gronwall’s Lemma, one can get(112)vε−v0T,∞≤CM,TIIVεT,∞≤CM,TIIVεγ.
Applying Lemma4 with
(113)Ft,x,s,z=KH∗Gt−s,x−z1s≤t,β0=12,CF=C,ρ=εM1+v0T,∞+θ,Nt,x=ε1vεT,∞<v0T,∞+θ,for any fixed θ>0, one can get that for any β,G≥εK1+v0∣T,∞+θCF1/2Çβ,β0Cβ,β0,
(114)PIIVε∣β,β≥M,vεT,∞<v0T,∞+θ≤Lexp−G2εM2CC2β,β01+v0∣T,∞+θ2.
We can get(115)limsupε→0e−2εlogPvε∣T,∞≥v0T,∞+θ≤limsupε→0e−2εlogPvε−v0∣T,∞≥θ=−∞.
Together with (73), (78), (82), (83), and (85), we have
(116)limsupε→0e−2εlogPIIIT,μ',θ'aε>δ≤limsupε→0e−2εlogPIIIT,μ',θ'2>εaεδCβ,T,m≤limsupε→0e−2εlogPIIIT,μ',θ'2>εaεδCβ,T,m,vεT,∞<v0T,∞+θ+PvεT,∞≥v0T,∞+θ≤limsupε→0−δεaεCβ,T,mm2CC2β,β01+v0∣T,∞+θ2∨limsupε→0e−2εlogPvε∣T,∞≥v0T,∞+θ=−∞.
This completes the proof.


## 3.1. Main Results

For any functionϕ defined on 0,T×D, let
(22)ϕt,∞=supϕs,z:s,z∈0,t×D,ϕt,γ=supϕs1,z1−ϕs2,z2s1−s2μ+z1−z2θγ:s1,z1≠s2,z2∈0,t×D,where μ∈0,α+1H−1/α, θ∈0,min1,α+1H−1, and γ∈0,1. Let
(23)ϕγ=ϕT,∞+ϕT,γ.LetCγ0,T×D be the functions ϕ:0,T×D which satisfy ϕγ<∞, endowed with the ϕγ−norm.Define(24)E=ex,t=∫0t∫0xe¯s,zdsdz,e¯∈L20,T×D,which is a Cameron-Martin space endowed with the norm
(25)eℰ=∫0t∫0xe¯2s,zdzds1/2.Supposee∈E. Now, let ve be the solution of the following deterministic equation:
(26)∂ve∂tt,x=Dδ,αvet,x+fvet,x+Fe¯t,x,ve0,x=0,where
(27)Fe¯x,t≔∂2∂x∂t∫0t∫0xKHt,se¯z,sdzdssatisfies the following:
(28)∫0t∫DGαδt−s,x−zFe¯z,edzds=∫0t∫DKH∗Gαδt−s,x−ze¯z,sdzds.The above (26) shows a unique mild solution.
(29)Zex,t=∫0t∫DGαδt−s,x−zfs,z,Zez,sdzds+∫0t∫DKH∗Gαδt−s,x−ze¯z,sdzds.Similar to Jiang et al. [5], one can get the following:Theorem 1.
LetH∈1/2,1. Under Assumption1, the law of the solution to (1) satisfies a deviation principle onCγD×0,Twith the good rate function:
(30)Iφ=inf12eℰ2;Ze=φ,with the conventioninf∅=∞, where0<γ<1.
More precisely, for any Borel measurable subset B ofCγ0,T×D,
(31)−infφ∈BoIφ≤liminfε→0εlogPZe∈B≤limsupε→0εlogPZe∈B≤−infφ∈B¯Iφ,whereBoandB¯denote the interior and the closure ofB, respectively.We furthermore suppose that the coefficients satisfy the following.Assumption 2.
f is differentiable, and the derivative f′ of f is Lipschitz. That is to say, there exist positive constant m and m′ which satisfy the following:
(32)f′y−f′x≤m′y−x,∀x,y∈D.Together with the Lipschitz off, we conclude that
(33)f′x≤m.Now, we give the following central limit theorem.Theorem 2.
Letfand its derivativef′satisfy Assumptions 1and2. Then, forp≥1,ve−v0/εconverges inLpto a random fieldUonCγ0,T×Dwith0<γ<1, determined by
(34)∂Ut,x∂t=Dδ,αUt,x+f′v0t,xUt,x+BHdt,dx,U0,x=0,for all,t,x∈0,T×D.Let the functionUe be the solution to the following partial differential equation:
(35)∂Uet,x∂t=Dδ,αUet,x+f′v0t,xUet,x+Fe¯t,x,Ue0,x=0.Under Assumptions1 and 2, by Theorem 1, one can get U/aε which satisfies large deviation principles on Cγ0,T×D with the speed e2ε and the good rate function satisfies the following:
(36)Iφinf12eℰ2,Ue=φ,+∞,otherwise.Now, the second result is given as follows:Theorem 3.
In moderate deviation principle, letH∈1/2,1. Under the Assumptions1and2, then, the random field1/εaεvε−v0satisfies a large deviation principle on the spaceCγ0,T×Dwith speeda2εand the good rate functionIφdefined by (36), where0<γ<1.


## 3.2. Convergence of the Solution

Lemma 2.
LetH∈1/2,1. Under Assumption1, then, there exists a unique solution to (1). Moreover, for anyp∈1,∞,T>0,
(37)supε≥1supt,x∈0,T×DEvεt,xp<∞.Asε→0, we get the convergence of vε as follows:Proposition 1.
LetH∈1/2,1andp≥2. By Assumption1, there exists a constantcm,p,H,T which satisfies the following:
(38)Evε−v0t,∞p≤εp/2cm,p,H,T.Proof.
Note that(39)vεt,x−v0t,x=∫0t∫DGαδt−s,x−zfvεs,z−fv0s,zdzds+ε∫0t∫DGαδt−s,x−zBHdsdz.One can get
(40)vεt,x−v0t,xT,∞p≤2p−1supt,x∈0,T×D∫0t∫DGαδt−s,x−zfvεs,z−fv0s,zdzdsp+εp/22p−1supt,x∈0,T×D∫0t∫DGαδt−s,x−zBHdsdzp.Set
(41)N1εt,x=∫0t∫DGαδt−s,x−zfvεs,z−fv0s,zdzds,N2εt,x=∫0t∫DGαδt−s,x−zBHdsdz.
Together with Hölder’s inequality, the Lipschitz condition (C) and (5) of LemmaA.1, for 1/α<q<α, we have
(42)EN1εT,∞p≤mpsupt,x∈0,T×DGαδt−s,x−zqdzdsp/q×E∫0Tvε−v0T,∞pdt≤Cp,m,TE∫0Tvε−v0T,∞pdt,where 1/p+1/q=1.
For any0≤t≤T, x,z∈D, p>2, by (21) and (A.9), there exist θ∈0,min1,α+1H−1 which satisfies the following:
(43)EN2εt,x−N2εt,x′p=E∫0t∫DGαδt−s,x−z−Gαδt−s,x′−zBHdsdzp≤∫0t∫DKH∗Gαδt−s,x−z−Gαδt−s,x′−z2dsdzp/2≤∫0t∫DGαδt−s,x−z−Gαδt−s,x′−z1/HdsdzHp≤mx−x′pθ.
Similarly,(44)EN2εt,x−N2εt′,xp=E∫0t∫DGαδt−s,x−z−Gαδt′−s,x−zBHdsdzp≤∫0t∫DKH∗Gαδt−s,x−z−Gαδt′−s,x−z2dsdzp/2≤Cp,m,T∫t't∫DKH∗Gαδt−s,x−z2dsdzp/2+Cp,m,T∫0t'∫DKH∗Gαδt−s,x−z−Gαδt'−s,x−z2dsdzp/2.
By (21), (A.10), and (A.11), for μ∈0,α+1H−1/α, one can get
(45)∫t't∫DKH∗Gαδt−s,x−z2dsdzp/2≤CH∫t't∫DGαδt−s,x−z1/HdsdzHp≤CHt−t′pα+1H−1/αand
(46)∫0t'∫DKH∗Gαδt−s,x−z−Gαδt′−s,x−z2dsdzp/2≤CH∫t't∫DKH∗Gαδt−s,x−z−Gαδt′−s,x−z1/HdsdzHp≤CHt−t′pμ.
Together with (43), (44), (45), and (46), one can get for any p≥2 and t′,x′,t,x∈0,T×D, there exists a constant β>0 such that
(47)EN2εt,x−N2εt,x′p≤Cp,m,H,Tx−x′θ+t−t′μpβ,where Cp,m,H,T is independent of ε. For p>2/β, by Garsis-Rodemich-Rumser’s Lemma, there exist a constant C and a random variable Mp,εω satisfying
(48)N2εt,x−N2εt,x′p≤mp,εωx−x′θ+t−t′μpβ−2logcx−x'θ+t−t'μ2and
(49)supε∈0,1Emp,ε<∞.
If we chooset′=0 in (48), we can get
(50)Esupt,x∈0,T×D∫0t∫DGαδt−s,x−zBHdsdzp≤cm,p,H,Tsupε∈0,1Emp,ε<∞.
By Gronwall’s inequality and (40), (42), and (48), there exists a constant Cp,m,H,T satisfying
(51)Evε−v0t,∞p≤εp/2cm,p,H,T.
The proof is completed.


## 3.3. The Proof of the Main Results

We first prove Theorem1. Our proof is based on the following proposition (see Doss and Priouret [28]).Proposition 2.
SupposeXjϵ,ϵ>0 and j=1,2are two families of random variables andEj,dj and j=1,2are two Polish spaces. Suppose that
(1)
There exists a mapK:I1<∞→E1such thatI1≤a→E1is continuous for anya<∞.
(2)
X1ϵ,ϵ>0satisfies with the rate functionsI1:E1→0,∞.
(3)
For anyR,δ,a>0, there existρ>0andϵ0>0satisfyingI1≤aandϵ≤ϵ0for allh∈E1, and
(52)Pd1X1ϵ≤ρ,d2x2ϵ,Ke≥δ≤exp−Rϵ2.
Then,X2ϵ,ϵ>0satisfies a LDP with the rate functions
(53)Ih=infI1e:Ke=h.To prove Theorem1, one only needs to prove
(i)
Under some topology,Ze:e:eE≤a→C0,T,LpD is continuous for any a>0.
(ii)
In Freidin-Wentzell inequality, for anyR>0, η>0, and e∈E, there exist a δ>0 satisfying
(54)limϵ→0supϵ2logPϵW−e∞<δ,supt∈0,Tuϵt,·−Zet,·∥2≥ρ≤−R.Theorem 4.
When the level seteE≤ais endowed with the topology of uniform convergence on0,T×D,
(55)Ze:e:eE≤a→C0,T,LpDis the continuous map for anya∈0,+∞.Proof.
One only needs to prove that for fixeda>0, e,h∈∥e∥E≤a,
(56)supt∈0,T∫ℝZet,x−Zht,x2dx≤Ce−hℰ.
Note that(57)Zet,x−Zht,x=∫0t∫ℝGαδt−s,x−zfs,z,Zes,z−fs,z,Zhs,zdzds+∫0t∫ℝKH∗Gαδt−s,x−ze¯s,zdzds≔It,x+IIt,x.
Using (A.5) in Appendix with p=q=2, then ρ=1, one can get
(58)It,x22≤C∫0tt−s−1/α+1/αρfs,z,Zes,z−fs,z,Zhs,z2ds≤Csup0≤s≤tZes,⋅−Zhs,⋅2.
Now, we deal withIIt,x, together with
(59)IIt,x22=∫ℝ∫0t∫ℝKH∗Gαδt−s,x−ze¯s,z−h¯s,zdzds2dx≤C∫ℝ∫0t∫ℝKH∗Gαδt−s,x−z2dzdsdx×e−hE2≔CMt×e−hℰ2.
RecallingKH∗ is defined by (11), one can get
(60)KH∗ϕs,x≤maxw∈s,Tϕw,xKHT,s,Mt=∫ℝ∫0t∫ℝKH∗Gαδt−s,x−z2dzdsdx≤∫ℝ∫0t∫ℝmaxw∈s,tGαδt−s,x−z2KH2t,sdzdxds≤C∫0tt−s−1/αt−s2H−1ds<∞.
Using Gronwall’s inequality, we can get(61)supt∈0,T∫ℝZet,x−Zht,x2dx≤Ce−hℰ.
The proof of the theorem is completed.We now prove the Freidin-Wentizell inequality as follows:SupposeWt,x,t,x∈0,T×ℝ is a Brownian sheet. For e∈E and ϵ>0, we define
(62)W~t,x=Wt,x−ex,tϵand
(63)d~pdp=exp1ϵ∫0T∫ℝe¯t,xWdt,dx−12ϵ2∫0T∫ℝe¯2t,xdxdt≔ZT.Using Girsanov’s theorem, the processW~ is a Brownian sheet under P~. Suppose v~εt,x is a solution of (1) under p~. Then,
(64)v~ϵt,x=∫ℝGαδt,x−zv0zdz+∫0t∫ℝGαδt−s,x−zft,x,v~εs,zdzds+∫0t∫ℝKH∗Gαδt−s,x−ze¯s,zdzds+ϵ∫0t∫ℝKH∗Gαδt−s,x−zW~ds,dz.Now, one can prove (36). Note that, under p~, then,
(65)u~ϵt,·−Zet,·2=∫0t∫ℝGαδt−s,·−zft,x,u~εs,z−ft,x,Zes,zdzds2+ϵ∫0t∫ℝKH∗Gαδt−s,·−yW~ds,dz2≤∫0t∫ℝu~εs,·−Zes,·2dzds+ϵ∫0t∫ℝKH∗Gαδt−s,·−zW~ds,dz2.So underp~, by Gronwall’s Lemma, one can get
(66)supt∈0,Tu~εs,·−Zhs,·2≤Csupt∈0,Tϵ∫0t∫ℝKH∗Gαδt−s,x−zW~ds,dz2.Now, one can change (34) the proof to the following theorem.Theorem 5.
Supposee∈Eande∈E≤a. For eachR>0,η>0, ande∈E, there exists a constantδ>0satisfying
(67)limϵ→0supϵ2logP~supt∈0,Tϵ∫0t∫ℝKH∗Gαδt−s,x−zW~ds,dz2≥ρ,ϵW~∞<δ≤−R.In the following, we give a key Lemma to prove Theorem1, which is similar to Candon-Weber [2], and the proof is omitted.Lemma 3.
SupposeF:0,T×D2→ℝ, CF>0andβ0>0satisfying
(68)∫0T∫DFt,x,v,z−Fs,y,v,z2dudz≤CFt−sμ+x−yθβ0,for any s,z and t,x∈0,T×D. Suppose N:Ω×0,T×0,1→ℝ which is an almost surely continuous,Ft-adapted process satisfyingsupNt,x:t,x∈0,T×D≤ρ,a,s, and fort,x∈0,T×D, suppose
(69)Ft,x=∫0T∫DFt,x,v,zNv,zWdvdz.
Then, for any0<β<β0/2, there exist a positive constant C, Ç(β,β0), andCβ,β0such that for allM≥ρCF1/2Ç(β,β0)Cβ,β0,
(70)PFt,β≥M≤Cexp−M2ρ2CzÇβ,β0C2β,β0.Proof of Theorem 5.
Suppose(71)Ft,x,v,z=ϵKH∗Gαδt−v,x−z.Then, there exists β0>0 satisfying
(72)∫0t∫DKH∗Gαδt−v,x−z−Gαδs−v,y−z2dvdz≤C∫0t∫ℝKH∗Gαδt−v,x−z−Gαδs−v,x−z2dvdz+∫0t∫ℝKH∗Gαδs−v,x−z−Gαδs−v,y−z2dvds≤C∫ot∫ℝGαδt−v,x−z−Gαδs−v,x−z1/Hdvds2H+∫0t∫ℝGαδs−v,x−z−Gαδs−v,y−z1/Hdvdz2H≤Ct−sμ+x−zθβ0.
IfR≤ρ2/2CFC2V and ρ≤KvCvCF, by Lemma 2, we can get
(73)P~supt∈0,Tϵ∫0t∫ℝKH∗Gαδt−s,x−zW~ds,dz2≥ρ≤2T2π2+1exp−Rϵ2.
The proof of Theorem3 is completed.In the following, we first give Garsis-Rodemich-Rumser’s Lemma in Bally et al. [29].Lemma 4.
Assumep>1andUεt,x:t,x∈0,T×Dare a family of real-valued stochastic processes. Suppose the following is true.Assumption 3.
One has(74)limε→0EUεt,xp=0as t,x∈0,T×D.Assumption 4.
For anyt,x,s,z∈0,T×D, there existsν>0satisfying
(75)EUεt,x−Uεs,zp≤Ct−s+x−z2γ.
Then, for anyr∈1,p,γ∈0,β/p,
(76)limε→0EUεγr=0.Now we can prove Theorem 2.Proof of Theorem 2.
SetUε=vε−v0/ε. We will prove that
(77)limε→0EUε−Uγr=0.
To this end, we need to prove that Assumptions3 and 4 are satisfied for Wε=Uε−U. Note that
(78)Uεt,x−Ut,x=∫0t∫DGαδt−s,x−zfvεs,z−fv0s,zε−f′v0s,zUs,zdsdz≕IIIεt,x+IVεt,xwhere
(79)IIIεt,x=∫0t∫DGαδt−s,x−zfvεs,z−fv0s,zε−f′v0s,zUεs,zdsdzand
(80)IVεt,x=∫0t∫DGαδt−s,x−zf′v0s,zUεs,z−Us,zdsdz.
Using Taylor’s formula, there exists aξεt,x such that
(81)fvεt,x−fv0t,x=f′v0t,x+ξεt,xvεt,x−v0t,x×vεt,x−v0t,x.
Note thatf′ is Lipschitz continuous and ξεt,x∈0,1; one can get
(82)f′v0t,x+ξεt,xvεt,x−v0t,x×vεt,x−v0t,x≤m′ξεt,xvεt,x−v0t,x≤m′vεt,x−v0t,x.
So(83)IIIεt,x≤m′∫0t∫DGαδt−s,x−zvεt,x−v0t,xUεs,zdsdz=εm′∫0t∫DGαδt−s,x−zUεs,z2dsdz.
Using Hölder’s inequality, forp<2 and 1/α<q<α, one can get
(84)EIIIε∞tp≤εp/2m′psupt,x∈0,T×D∫0t∫DGαδt−s,x−zqdsdzp/q×∫0tEUε∞s2pds,where 1/p+1/q=1. Together with (5) of Lemma A.1 and Proposition 1, there exists a constant Cp,m,m′,T only depending on p,m,m′,T satisfying
(85)EIIIε∞tp≤εp/2Cp,M,M',T.
Sincef′≤M, for p>2, together with Hölder’s inequality and (5) of Lemma A.1, we can get
(86)EIVε∞tp≤Mpsupt,x∈0,T×D∫0t∫DGαδt−s,x−zqdsdzp/q×E∫0tUε−U∞spds,where 1/p+1/q=1. Together with (52), (56), and (57), we can get
(87)EUε−U∞tp≤Cp,m,m',Tεp/2+E∫0tUε−U∞spds.
Using Gronwall’s inequality, one can get(88)EUε−U∞tp≤εp/2Cp,m,m',T→0,ε→0,which implies Assumption 3 in Lemma 4.
Now we prove Assumption4 in Lemma 4. We will prove IIIε and IVε satisfy Assumption 4 in Lemma 4. Using Hölder’s inequality and (A.9), one can get
(89)EIIIεt,x−IIIεt,x'p≤Eεm'∫0t∫DGαδt−s,x−z−Gαδt−s,x'−zUεs,z2dsdzp≤εp/2M'p∫0t∫DGαδt−s,x−z−Gαδt−s,x'‐zqdsdzp/q×∫0t∫DEYεs,z∞s2pdsdz1/p≤εp/2Cp,m,m',Tx−x'pθ',where θ′∈0,min1,α+1/q−1 and 1/p+1/q=1. Similarly,
(90)EIVεt,x−IVεt',xp≤Eεm'∫0t∫DGαδt−s,x−z−Gαδt'‐s,x‐zUεs,z2dsdzp≤εp/2Cp,m,T∫0t∫DGαδt−s,x−z−Gαδt'‐s,x‐zUεs,z2dsdzp≤εp/2Cp,m,T∫t't∫DGαδt−s,x−z−Gαδt'‐s,x‐zUεs,z2dsdzp+εp/2Cp,m,T∫0t'∫DGαδt−s,x−zUεs,z2dsdzp≤Cp,m,m',Tt−t'pμ,where μ'∈0,α+1/q−1/α. Together with (89) and (90), we can get
(91)EIIIεt,x−IIIεs,zp≤Ct−s+x−z2β
Similarly, one can get(92)EIVεt,x−IVεs,zp≤Ct−s+x−z2β.
Together with (91) and (92), we can get
(93)EUεt,x−Ut,x−Uεs,z−Us,zp≤Ct−s+x−z2β.
Then, for anyp>2 and q∈1,α such that γ∈0,β/p and r∈1,p, using Lemma 4,
(94)limε→0EVεγr=0.
The proof of Theorem2 is completed.Proof of Theorem 3.
By Theorem1, U/aε obeys large deviation principles on Cγ0,T×D, with the rate function I given by (30) and the speed function h2ε. Using Dembo and Zeitouni ([30] Theorem 4.2.13), to prove the large deviation principles of Uε/aε is e2ε-exponentially equivalent to U/aε, that is,
(95)limsupε→0e−2εlogPUε−Uμ′,θ′aε>δ=−∞holds for any δ>0.
Since(96)Uε−Uγ≤CT,µ′,θ′Uε−UT,γ.
To prove (67), we only need to prove
(97)limsupε→0e−2εlogPUε−Ut,μ′,θ′aε>δ=−∞.
Note the decomposition(98)Uεt,x−Ut,x=IIIεt,x+IVεt,x.
For anyq∈1,α,x,x′∈0,1, 1/p+1/q=1, and 0≤s≤t≤T, by Hölder’s inequality, (32), and (A.9), we can get
(99)IVεt,x−IVεt,x′≤m′∫0t∫DGαδt−s,x−z−Gαδt−s,x′−zUεs,z−Us,zdsdz≤m′∫0t∫DGαδt−s,x−z−Gαδt−s,x′−zqdsdz1/q×∫0t∫DUεs,z−Us,zpdsdz1/p≤M′x−x′θ×∫0tUε−U∞up1/p.where θ′∈0,min1,α+1/q−1. Similarly,
(100)IVεt,x−IVεt′,x≤m′∫0t'∫DGαδt−s,x−z−Gαδt'‐s,x‐zUεs,z−Us,zdsdz≤m′∫0t∫DGαδt−s,x−z−Gαδt'−s,x−zqdsdz1/q×∫0t'∫DUεs,z−Us,zpdsdz1/p+m′∫t't∫DGαδt−s,x−zqdsdz1/q×∫t't∫DUεs,z−Us,zpdsdz1/p≤m′t−t′μ′×∫0tUε−U∞up1/p.where μ′∈0,α+1/q−1/α. Together with (99) and (100), we can get
(101)EIVεt,x−IVεs,zp≤Ct−sµ′+x−zθ′.
Note thatUε−Us,∞≤Cs,µ′,θ′Uε−Us,γ, we have
(102)IVεt,xt,γ≤C∫Cs,μ′,θ′Uε−Us,γpds.
Thus, fort∈0,T, we have
(103)Uε−Ut,γp≤Cp,m,TIIIεt,γ+∫0tUε−Us,γpds.
Applying Gronwall’s Lemma toft=Uε−Ut,γp, we have
(104)Uε−UT,γp≤Cp,T,mIIIεT,γp.
By (64) to prove (67), we only need to prove that for any δ>0,
(105)limsupε→0e−2εlogPIεT,γaε≥δ=−∞.
Note that(106)IIIεt,x=∫0t∫DGαδt−s,x−zAεs,zdsdz,where
(107)Aεs,z=fvεs,z−fv0s,zε−f′v0s,zUεs,z.
By the same method in the proof of (59), we have
(108)IIIεT,γ≤CT,µ′,θ′AεT,∞.
Similar to the proof of (53), we have
(109)AεT,∞≤m′vε−v0T,∞2ε.
Together with (40) and (A.11), for any t∈0,T, we can get that
(110)vε−v0t,∞≤supt,x∈0.T×D∫0l∫DGαδl−s,x−zmvεs,z−v0s,zdsdzsupt,x∈0,T×Dε∫0l∫DGαδt−s,x−zBHdsdz≤Cm,T∫0tvε−v0s,∞ds+IIVεt,∞,where
(111)IIVεt,x=ε∫01∫DGαδt−s,x−zBHdsdz=ε∫01∫DKH∗Gαδt−s,x−zWdsdz.
By Gronwall’s Lemma, one can get(112)vε−v0T,∞≤CM,TIIVεT,∞≤CM,TIIVεγ.
Applying Lemma4 with
(113)Ft,x,s,z=KH∗Gt−s,x−z1s≤t,β0=12,CF=C,ρ=εM1+v0T,∞+θ,Nt,x=ε1vεT,∞<v0T,∞+θ,for any fixed θ>0, one can get that for any β,G≥εK1+v0∣T,∞+θCF1/2Çβ,β0Cβ,β0,
(114)PIIVε∣β,β≥M,vεT,∞<v0T,∞+θ≤Lexp−G2εM2CC2β,β01+v0∣T,∞+θ2.
We can get(115)limsupε→0e−2εlogPvε∣T,∞≥v0T,∞+θ≤limsupε→0e−2εlogPvε−v0∣T,∞≥θ=−∞.
Together with (73), (78), (82), (83), and (85), we have
(116)limsupε→0e−2εlogPIIIT,μ',θ'aε>δ≤limsupε→0e−2εlogPIIIT,μ',θ'2>εaεδCβ,T,m≤limsupε→0e−2εlogPIIIT,μ',θ'2>εaεδCβ,T,m,vεT,∞<v0T,∞+θ+PvεT,∞≥v0T,∞+θ≤limsupε→0−δεaεCβ,T,mm2CC2β,β01+v0∣T,∞+θ2∨limsupε→0e−2εlogPvε∣T,∞≥v0T,∞+θ=−∞.
This completes the proof.



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*Source: 7402764-2018-07-10.xml*