# Stochastic Fractional Heat Equations Driven by Fractional Noises

Authors: Xichao Sun; Ming Li
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper is concerned with the following stochastically fractional heat equation on(t,x)∈[0,T]×Rd driven by fractional noise: ∂u(t,x)/∂t=Dδαu(t,x)+WH(t,x)⋄u(t,x), where the Hurst parameter H=(h0,h1,…,hd) and ⋄ denotes the Skorokhod integral. A unique solution of that equation in an appropriate Hilbert space is constructed. Moreover, the Lyapunov exponent of the solution is estimated, and the Hölder continuity of the solution on both space and time parameters is discussed. On the other hand, the absolute continuity of the solution is also obtained.


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## Body

## 1. Introduction

Recently, there has been intense interest (see, e.g., Podlubny [1], Samko et al. [2], Heydari et al. [3, 4], Cattani et al. [5, 6], Liao [7], and Hu [8]) in fractional calculus and its applications. Many mathematical problems in physics and engineering with respect to systems and processes are represented by a kind of equations, more precisely fractional order differential equations driven by fractional order noise. One of the emerging branches of this study is the theory and applications of fractional (partial) differential equations. The increasing interest in this class of equations is motivated both by their applications to fluid dynamic traffic model, viscoelasticity, heat conduction in materials with memory, electrodynamics with memory and also because they can be employed to approach nonlinear conservation laws (see [9] and references therein). For instance, the research discussed by Gurtin and Pipkin [10] and Nunziato [11] provides a description of heat conduction in materials with fading memory. Besides, noise or stochastic perturbation is unavoidable and omnipresent in nature as well as in man-made systems. Therefore, it is of great significance to import the stochastic effects into the investigation of fractional (partial) differential systems.As known, publications treating stochastic partial differential equation (SPDE) involving fractional derivatives gain interests. In fact, these kinds of equations may be widely used in physics, fractal medium, quantum fields, risk management, statistical mechanics, and other areas (see Droniou and Imbert [12], Uchaikin and Zolotarev [13], Toma [14], Bakhoum and Toma [15], and Li et al. [16, 17]). Most of them investigate evolution type equations driven by a fractional power of the Laplacian. Mueller [18] and Wu [19] proved the existence of a solution of stochastically fractional heat and Burgers equation perturbed by a stable noise, respectively. Boulanba et al. [20] studied the existence, uniqueness, Hölder regularity, and absolute continuity of the solution for a class of fractional stochastic partial differential equations driven by spatially correlated noise. Other related references include Chang and Lee [21], Debbi and Dozzi [22], Sun et al. [23], Truman and Wu [24], Liu et al. [25], and Wu [26].On the other hand, there has been an increasing interest in studying SPDEs driven by fractional noise. Recall that a fractional Brownian motion (fBm) is a centered Gaussian process(Bt)t≥0 with the covariance given by (1)E(BtBs)=RH(t,s)≔12t2H+s2H-t-s2H,with H∈(0,1), referring Biagini et al. [27], Mishura [28], and Nualart [29] for a comprehensive account on the fBm. The fBm has some interesting properties, such as self-similarity, Hölder continuity, and long-range dependence. It has been applied in various scientific areas including telecommunications, turbulence, image processing, and finance engineering. Moreover, there have been many papers about SPDEs driven by fractional noise. Hu [30] showed the existence and uniqueness of the solutions for a class of second-order stochastic heat equations, via chaos expansion. Nualart and Ouknine [31] explored the existence and uniqueness of mild solution to a class of second-order heat equations with additive fractional noise (fractional in time and white in space) when the Hurst parameter H>1/2. In a successive paper, Tindel et al. [32] studied a linear stochastic evolution equation driven by an infinite-dimensional fBm in the cases of the Hurst parameter above and below 1/2, respectively. Hu and Nualart [33] studied the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of an fBm with H∈(0,1) in time. More works for the fields can be found in Balan [34], Balan and Tudor [35, 36], Bo et al. [37, 38], Jiang et al. [39–41], and the references therein.LetBH(t,x), (t,x)∈[0,T]×Rd, be the canonical fractional Brownian fields (FBF) with the Hurst parameter H on the canonical probability space (Ω,F,P), where H=(h0,h1,…,hd) with 1/2<hi<1, 0≤i≤d. The fractional white noise is denoted by WH(t,x); that is, formally WH(t,x)=∂d+1/∂t∂x1∂x2⋯∂xdBH(t,x). In this paper, we focus on the following stochastically fractional partial differential equation (t,x)∈[0,T]×Rd driven by fractional noise:(2)∂u(t,x)∂t=Dδαu(t,x)+WH(t,x)⋄u(t,x),where “⋄” denotes the Skorokhod integral, (t,x)∈[0,T]×Rd, d≥1, α=(α1,…,αd), δ=(δ1,…,δd), and Dδα denotes a nonlocal fractional differential operator defined by (3)Dδα=∑i=1d‍Dδiαi,where Dδiαi denotes the fractional differential derivative with respect to the ith coordinate defined via its Fourier transform F by (4)F(Dδαφ)(ξ)=-ξαexp⁡-iδπ2sgnξF(φ)(ξ),where i2+1=0. The precise meaning of the solution of (2) will be stated in Section 2.The structure of this paper is as follows. In Section2, we briefly present some basic notations and preliminaries. Section 3 consists of the existence and the Lyapunov exponent estimate of the solution to (2). In Section 4, we check the Hölder continuity of the solution {u(t,x),(t,x)∈[0,T]×Rd} with both space and time parameters. In Section 5, we prove that the law of the solution of (2) is absolutely continuous with respect to the Lebesgue measure on Rd. Finally, Section 6 concludes the paper.


## 2. Preliminaries

In this section, we will first introduce multiple stochastic integrals with respect to fractional Brownian fields and define a solution of (2) in Sρ sense after proposing the fractional differential operator Dδα and introducing some properties. Then, we will recall the Malliavin calculus with respect to fractional noises.


### 2.1. Skorokhod Integral with respect to Fractional Brownian Fields

Definition 1.
A multiparameter fractional Brownian fieldBH={BH(t,x),(t,x)∈[0,T]×Rd} with multiparameter H=(h0,h1,…,hd) for hi∈(0,1) and i∈{0,1,2,…,d} is a centered Gaussian field defined on some probability space (Ω,F,P) with the covariance as follows:(5)EBHt,xBHs,y=12d+1t2h0+s2h0-t-s2h0×∏i=1dxi2hi+yi2hi-xi-yi2hifor all t,s∈[0,T], x=(x1,…,xd), and y=(y1,…,yd)∈Rd.Throughout the paper, we restrict our consideration on the multiparameter fractional Brownian field with the parameterH=(h0,h1,…,hd) for hi∈(1/2,1).Firstly, we briefly introduce the stochastic integral with respect to the fractional Brownian fieldBH(t,x).For anyx=(x1,x2,…,xn) and y=(y1,y2,…,yn), we write φH(x,y)≔∏i=1dφhi(xi,yi), where φh≔h(2h-1)x-y2h-2. Introduce the following Hilbert space: (6)H1≔∫[0,T]2‍f:[0,T]×Rd⟶R;hhhhhhfH2=∫[0,T]2‍∫R2d‍φh0s-rφHx,yfs,xhhhhhhhhhhhhhhhhhhhhhh×fr,ydxdydsdr<∞∫[0,T]2‍,where dx=dx1dx2⋯dxd and dy=dy1dy2⋯dyd. Let f∈H1; one can define the following stochastic integral:(7)∫0t‍∫Rd‍fs,xdBHt,x;t∈0,T(see, e.g., Hu [30]).Proposition 2.
Letf,g∈H1. Then, (1)
E[∫0T‍∫Rd‍f(s,x)dBH(s,x)]=0.(2)
E[∫0T‍∫Rd‍f(s,x)dBH(s,x)∫0T‍∫Rd‍g(s,x)dBH(s,x)]=〈f,g〉H1.Denote by⊗ the symmetric tensor product. Let {e1,e2,…} be an orthonormal basis of H1. Then, {ei1⊗ei2⋯⊗ein,1≤i1,i2,…,in<∞} is an orthonormal basis of Hn≔H1⊗n. It is easy to see that (8)Hn≔f:0,Tn×Rnd⟶R;hhhhhh∫[0,T]2n‍∫R2nd‍∏i=1nφh0si-riφHui,vifs,uhhhhhhhhhhhhhhh×fr,vdsdrdu1dv1⋯dundvnhhhhhhhh<∞f:0,Tn×Rnd⟶R;,where we denote (9)u=(u1,u2,…,un),v=(v1,v2,…,vn).Define(10)Hn≔f∈Hn;f((s1,u1),…,(sn,un))hhhhhhh=f((sσ1,uσ1),…,(sσn,uσn))hhhhhhhforallpermutationshhhhhhhσ=σ1,…,σnof1,2,…,nf∈Hn;fs1,u1,…,sn,un.We call f∈Hn symmetric to n ((d+1)-dimensional) variables (s1,u1),…,(sn,un). Now, define (11)Dn:=f∈Hn;f=Σai1,…,inei1⊗ei2⋯⊗ein,ai1,…,in∈R.Denote by ⊗ the symmetric tensor product. Let Hn(x) be the Hermite polynomial of degree n≥0. It is defined by (12)Hn(x)=-1nex2/2dndxne-x2/2,x∈R.For e∈H1 and eH1=1, define the multiple integral of Itö type of the function e⊗n by(13)∫[0,T]n‍∫Rnd‍e⊗n(s1,…,sn,u1,…,un)hhhhhhhh·dBH(s1,u1)⋯dBH(sn,un)=Hn(∫[0,T]‍∫Rd‍e(s,u)dBH(s,u)).Then,(14)Inei1⊗ei2⋯⊗ein=∫[0,T]n‍∫Rnd‍e⊗n(s1,…,sn,u1,…,un)hhhhhhi·dBH(s1,u1)⋯dBH(sn,un),by the polarization argument.For eachf∈Dn, we write(15)In(f)=∑‍ai1,…,inIn(ei1⊗ei2⋯⊗ein).Then, the following isometry holds:(16)EInf2=n!fHn2.Note that, for f∈Hn, there exists a sequence {fk}k∈N⊂Dn such that fn→f in Dn. It follows from (15) that {In(fk)}k∈N is Cauchy in L2(Ω), and the limit point of In(fk) (as k→∞) is independent of the choice of f∈Hn. We call the limit point the multiple integral of Itö type and denote it by(17)Inf≔∫[0,T]n‍∫Rnd‍f(s,u)dBH(s1,u1)⋯dBH(sn,un)=limk→∞In(fk),inL2(Ω)sense.It is easy to see that, for f,g∈Hn,(18)E[In(f)In(g)]=n!f,gHn.LetF=⨁n=0∞Fn, where Fn is the nth chaos of F  (see, e.g., Hu [42]). For each ρ∈R, we introduce a Hilbert space denoted by (19)Sρ={F=⨁n=0∞Fn;∑n=0∞‍n!ρEFn2<∞}and define (20)Fρ≔∑n=0∞‍n!ρEFn2.In particular, if ρ=0, one has Sρ=L2(Ω).


### 2.2. Definition of the Solution

In order to define the solution of (2), we will introduce the nonlocal fractional differential operator defined by (21)Dδα=∑i=1d‍Dδiαi,where α=(α1,…,αd), δ=(δ1,…,δd), and Dδiαi denotes the fractional differential derivative with respect to the ith coordinate defined via its Fourier transform F by (22)F(Dδαφ)(ξ)=-ξαexp⁡(-iδπ2sgn(ξ))F(φ)(ξ).In this paper, we will assume that |δi|≤min⁡{αi-[αi]2,2+[αi]2-αi}, i=1,…,d,[αi]2, is the largest even integer less or equal to αi (even part of αi), and δi∈2N+1.In one space dimension, the operatorDδα is a closed, densely defined operator on L2(R). It is the infinitesimal generator of a semigroup which is in general not symmetric and not a contraction. This operator is a generalization of various well-known operators, such as the Laplacian operator when α=2, the inverse of the generalized Riesz-Feller potential if α>2, and the Riemann-Liouville differential operator when |δ|=2+[α]2 or |δ|=α-[α]. It is self-adjoint only when δ=0 and, in this case, it coincides with the fractional power of the Laplacian, citing Debbi [43], Debbi and Dozzi [22], and Komatsu [44] for more details about this operator.According to Komatsu [44], Dδα can be represented for 1<α<2 by(23)Dδαφx=∫R‍φx+y-φx-yφ′xy1+αii×κ-δ1(-∞,0)(y)+κ+δ1(-0,+∞)(y)dyand for 0<α<1 by (24)Dδαφx=∫R‍φx+y-φxy1+αii×κ-δ1(-∞,0)(y)+κ+δ1(-0,+∞)(y)dy,where κ-δ and κ+δ are two nonnegative constants satisfying κ-δ+κ+δ>0 and φ is a smooth function for which the integral exists and φ′ is its derivative. This representation identifies it as the infinitesimal generator for a nonsymmetric α-stable Lévy process.LetGα,δ(t,x) be the fundamental solution of the following Cauchy problem:(25)∂u∂t(t,x)=Dδα(t,x),u(0,x)=δ0(x),t>0,x∈R,where δ0(·) is the Dirac distribution. By the Fourier transform, we see that Dα,δ(t,x) is given by F as follows:(26)Gα,δ(t,x)=12π∫R‍exp⁡-izx-tzαexp⁡-iδπ2sgnzdz.The relevant parameters α called the index of stability and δ (related to the asymmetry) improperly referred to as the skewness are real numbers satisfying |δ|≤min⁡{α-[α]2,2+[α]2-α}, and δ=0 when δ∈2N+1.Let us list some known results onGα,δ(t,x) that will be used later on (see, e.g., Debbi [43] and Debbi and Dozzi [22]).Lemma 3.
Letα∈(1,∞)/{N}; we have the following. (1)
The functionGα,δ(t,·) is not in general symmetric relatively to x and it is not everywhere positive.(2)
For anys,t∈(0,∞) and x∈R, (27)∂n∂xnGα,δt,x=s-n+1/αGα,δ(s-1t,s-1/αx)or equivalently (28)∂n∂xnGα,δ(t,x)=t-n+1/αGα,δ(1,t-1/αx).(3)
Gα(s,·)∗Gα,δ(t,·)=Gα,δ(s+t,·) for any s,t∈(0,∞).(4)
Forn≥1, there exist some constants C and Cn>0 such that, for all x∈R, (29)Gα,δ1,x≤C11+x1+α,∂n∂xnGα,δ1,x≤Cnxα+n-11+xα+n2.(5)
∫0T‍∫R‍|Gα,δt,x|λdtdx<∞ if and only if 1/α<λ<α.Ford≥1 and any multi-index α=(α1,…,αd) and δ=(δ1,…,δd), denote by Gα,δ(t,x) the Green function of the deterministic equation as follows:(30)∂u∂tt,x=Dδαt,x,u0,x=δ0x,t>0,x∈Rd.Clearly, (31)Gα,δt,x=∏i=1d‍Gαi,δi(t,xi)=12πd∫Rd‍exp⁡⁡t∑i=1d‍-iξ,x-t∑i=1d‍ξiαihhhhhhhhhhhhhh×exp⁡-iδπ2sgnξit∑i=1d‍dξ.Now, we describe a solution of (2) in Sρ sense.Definition 4.
We say that a stochastic field{u(t,x),(t,x)∈[0,T]×Rd} is a solution of (2) in Sρ sense, if (1)
u:R+×R×Ω is jointly measurable;(2)
∫0t‍∫Rd‍Gα,δ(t-s,x-z)u(s,z)dBH(s,z) is well defined for all t∈R+ and x∈Rd as an element of Sρ for certain ρ∈R;(3)
the following equation holds inSρ: (32)ut,x=Gα,δ(t)∗u(0,x)+∫0t∫Rd‍Gα,δ(t-s,x-z)u(s,z)dBH(s,z).The following embedding proposition given by Mémin et al. [45] is useful for our derivations below.Lemma 5.
Ifh∈(1/2,1) and f,g∈L1/h([a,b]), one has (33)∫ab‍∫ab‍fugvu-v2h-2dudv≤ChfL1/h([a,b])gL1/h([a,b]),where Ch>0 is a constant depending only on h.


### 2.3. Malliavin Calculus

In this subsection, let us brief the basic elements of the Malliavin calculus (see, e.g., Nualart [29]).SinceBH(t,x), (t,x)∈[0,T]×Rd, is Gaussian, we might develop the Malliavin calculus with respect to fractional noises in order to prove the existence of the laws of the solutions of SPDE driven by fractional noises.LetBH(φ)=∫0T‍∫Rd‍φ(t,x)BH(dt,dx) for φ∈H1 and (34)S=F=fBHφ1,…,BHφn;hf∈Cb∞(Rn),φi∈H1,n≥1F=fBH(φ1),…,BH(φn);be the space of all “smooth cylindrical” random variables, where Cb∞(Rn) denotes the class of all bounded infinitely differentiable functions on Rn, whose partial derivatives are also bounded. The Malliavin derivative of an element F=f(BHφ1,…,BHφn), with respect to BH, is defined by (35)Dt,xF≔∑i=1n‍∂f∂xiBHφ1,…,BHφnφi(t,x).Let D1,2 be the completion of S under the norm (36)F1,22≔EF2+EDFH12.Then, D1,2 is the domain of the closed operator D. For each h∈H1 and F∈S, define (37)DhF≔limɛ→0ddhfBH(h1)+ɛh1,hH1,…,BH(hn)hhhhhhh+ɛhn,hH1,which might be extended as a closed operator on L2(Ω) with the domain Dh being the closure of S under the norm(38)Fh2=E[F2+DhF2].Let {hn,n≥1} be an orthonormal basis of H1. Then, F∈D1,2 if and only if F∈Dhn for each n∈N and(39)∑n=1∞‍EDhnF2<∞.In this case, (40)DhF=DF,hH1.On the other hand, the divergence operatorδ:Domδ⊂L2(Ω;H1)→L2(Ω) is the adjoint of the operator D and is uniquely defined by the following duality relationship: u∈Domδ if and only if(41)E(Fδ(u))=EDF,uH1,∀F∈S.Note that u∈Domδ if and only if u is integrable with respect to BH. In the literature, δ is called the Skorokhod integral with respect to BH.The following propositions (see Wei [46] for the case of fractional noises), which are the Malliavin calculus with respect to fractional-colored noises, can deduce the laws for solutions to the corresponding stochastic partial differential equations.Proposition 6.
LetA∈F. If F is a square integrable random variable that is measurable with respect to the σ-field FAc, then (42)DF1A=0,a.s.Remark 7.
Let{u(t,x),(t,x)∈[0,T]×Rd} be an {Ft,t∈[0,T]}-adapted random field. By Proposition 6, we have Ds,yu(t,x)=0, a.s., for any 0≤t<s≤T and x,y∈Rd.Proposition 8.
SupposeF∈D1,2; if DFH12>0, a.s., then the distribution of the random variable F is absolutely continuous with respect to Lebesgue measure.Remark 9.
Propositions6 and 8 can be proved similarly to those in Nualart [29].


## 2.1. Skorokhod Integral with respect to Fractional Brownian Fields

Definition 1.
A multiparameter fractional Brownian fieldBH={BH(t,x),(t,x)∈[0,T]×Rd} with multiparameter H=(h0,h1,…,hd) for hi∈(0,1) and i∈{0,1,2,…,d} is a centered Gaussian field defined on some probability space (Ω,F,P) with the covariance as follows:(5)EBHt,xBHs,y=12d+1t2h0+s2h0-t-s2h0×∏i=1dxi2hi+yi2hi-xi-yi2hifor all t,s∈[0,T], x=(x1,…,xd), and y=(y1,…,yd)∈Rd.Throughout the paper, we restrict our consideration on the multiparameter fractional Brownian field with the parameterH=(h0,h1,…,hd) for hi∈(1/2,1).Firstly, we briefly introduce the stochastic integral with respect to the fractional Brownian fieldBH(t,x).For anyx=(x1,x2,…,xn) and y=(y1,y2,…,yn), we write φH(x,y)≔∏i=1dφhi(xi,yi), where φh≔h(2h-1)x-y2h-2. Introduce the following Hilbert space: (6)H1≔∫[0,T]2‍f:[0,T]×Rd⟶R;hhhhhhfH2=∫[0,T]2‍∫R2d‍φh0s-rφHx,yfs,xhhhhhhhhhhhhhhhhhhhhhh×fr,ydxdydsdr<∞∫[0,T]2‍,where dx=dx1dx2⋯dxd and dy=dy1dy2⋯dyd. Let f∈H1; one can define the following stochastic integral:(7)∫0t‍∫Rd‍fs,xdBHt,x;t∈0,T(see, e.g., Hu [30]).Proposition 2.
Letf,g∈H1. Then, (1)
E[∫0T‍∫Rd‍f(s,x)dBH(s,x)]=0.(2)
E[∫0T‍∫Rd‍f(s,x)dBH(s,x)∫0T‍∫Rd‍g(s,x)dBH(s,x)]=〈f,g〉H1.Denote by⊗ the symmetric tensor product. Let {e1,e2,…} be an orthonormal basis of H1. Then, {ei1⊗ei2⋯⊗ein,1≤i1,i2,…,in<∞} is an orthonormal basis of Hn≔H1⊗n. It is easy to see that (8)Hn≔f:0,Tn×Rnd⟶R;hhhhhh∫[0,T]2n‍∫R2nd‍∏i=1nφh0si-riφHui,vifs,uhhhhhhhhhhhhhhh×fr,vdsdrdu1dv1⋯dundvnhhhhhhhh<∞f:0,Tn×Rnd⟶R;,where we denote (9)u=(u1,u2,…,un),v=(v1,v2,…,vn).Define(10)Hn≔f∈Hn;f((s1,u1),…,(sn,un))hhhhhhh=f((sσ1,uσ1),…,(sσn,uσn))hhhhhhhforallpermutationshhhhhhhσ=σ1,…,σnof1,2,…,nf∈Hn;fs1,u1,…,sn,un.We call f∈Hn symmetric to n ((d+1)-dimensional) variables (s1,u1),…,(sn,un). Now, define (11)Dn:=f∈Hn;f=Σai1,…,inei1⊗ei2⋯⊗ein,ai1,…,in∈R.Denote by ⊗ the symmetric tensor product. Let Hn(x) be the Hermite polynomial of degree n≥0. It is defined by (12)Hn(x)=-1nex2/2dndxne-x2/2,x∈R.For e∈H1 and eH1=1, define the multiple integral of Itö type of the function e⊗n by(13)∫[0,T]n‍∫Rnd‍e⊗n(s1,…,sn,u1,…,un)hhhhhhhh·dBH(s1,u1)⋯dBH(sn,un)=Hn(∫[0,T]‍∫Rd‍e(s,u)dBH(s,u)).Then,(14)Inei1⊗ei2⋯⊗ein=∫[0,T]n‍∫Rnd‍e⊗n(s1,…,sn,u1,…,un)hhhhhhi·dBH(s1,u1)⋯dBH(sn,un),by the polarization argument.For eachf∈Dn, we write(15)In(f)=∑‍ai1,…,inIn(ei1⊗ei2⋯⊗ein).Then, the following isometry holds:(16)EInf2=n!fHn2.Note that, for f∈Hn, there exists a sequence {fk}k∈N⊂Dn such that fn→f in Dn. It follows from (15) that {In(fk)}k∈N is Cauchy in L2(Ω), and the limit point of In(fk) (as k→∞) is independent of the choice of f∈Hn. We call the limit point the multiple integral of Itö type and denote it by(17)Inf≔∫[0,T]n‍∫Rnd‍f(s,u)dBH(s1,u1)⋯dBH(sn,un)=limk→∞In(fk),inL2(Ω)sense.It is easy to see that, for f,g∈Hn,(18)E[In(f)In(g)]=n!f,gHn.LetF=⨁n=0∞Fn, where Fn is the nth chaos of F  (see, e.g., Hu [42]). For each ρ∈R, we introduce a Hilbert space denoted by (19)Sρ={F=⨁n=0∞Fn;∑n=0∞‍n!ρEFn2<∞}and define (20)Fρ≔∑n=0∞‍n!ρEFn2.In particular, if ρ=0, one has Sρ=L2(Ω).


## 2.2. Definition of the Solution

In order to define the solution of (2), we will introduce the nonlocal fractional differential operator defined by (21)Dδα=∑i=1d‍Dδiαi,where α=(α1,…,αd), δ=(δ1,…,δd), and Dδiαi denotes the fractional differential derivative with respect to the ith coordinate defined via its Fourier transform F by (22)F(Dδαφ)(ξ)=-ξαexp⁡(-iδπ2sgn(ξ))F(φ)(ξ).In this paper, we will assume that |δi|≤min⁡{αi-[αi]2,2+[αi]2-αi}, i=1,…,d,[αi]2, is the largest even integer less or equal to αi (even part of αi), and δi∈2N+1.In one space dimension, the operatorDδα is a closed, densely defined operator on L2(R). It is the infinitesimal generator of a semigroup which is in general not symmetric and not a contraction. This operator is a generalization of various well-known operators, such as the Laplacian operator when α=2, the inverse of the generalized Riesz-Feller potential if α>2, and the Riemann-Liouville differential operator when |δ|=2+[α]2 or |δ|=α-[α]. It is self-adjoint only when δ=0 and, in this case, it coincides with the fractional power of the Laplacian, citing Debbi [43], Debbi and Dozzi [22], and Komatsu [44] for more details about this operator.According to Komatsu [44], Dδα can be represented for 1<α<2 by(23)Dδαφx=∫R‍φx+y-φx-yφ′xy1+αii×κ-δ1(-∞,0)(y)+κ+δ1(-0,+∞)(y)dyand for 0<α<1 by (24)Dδαφx=∫R‍φx+y-φxy1+αii×κ-δ1(-∞,0)(y)+κ+δ1(-0,+∞)(y)dy,where κ-δ and κ+δ are two nonnegative constants satisfying κ-δ+κ+δ>0 and φ is a smooth function for which the integral exists and φ′ is its derivative. This representation identifies it as the infinitesimal generator for a nonsymmetric α-stable Lévy process.LetGα,δ(t,x) be the fundamental solution of the following Cauchy problem:(25)∂u∂t(t,x)=Dδα(t,x),u(0,x)=δ0(x),t>0,x∈R,where δ0(·) is the Dirac distribution. By the Fourier transform, we see that Dα,δ(t,x) is given by F as follows:(26)Gα,δ(t,x)=12π∫R‍exp⁡-izx-tzαexp⁡-iδπ2sgnzdz.The relevant parameters α called the index of stability and δ (related to the asymmetry) improperly referred to as the skewness are real numbers satisfying |δ|≤min⁡{α-[α]2,2+[α]2-α}, and δ=0 when δ∈2N+1.Let us list some known results onGα,δ(t,x) that will be used later on (see, e.g., Debbi [43] and Debbi and Dozzi [22]).Lemma 3.
Letα∈(1,∞)/{N}; we have the following. (1)
The functionGα,δ(t,·) is not in general symmetric relatively to x and it is not everywhere positive.(2)
For anys,t∈(0,∞) and x∈R, (27)∂n∂xnGα,δt,x=s-n+1/αGα,δ(s-1t,s-1/αx)or equivalently (28)∂n∂xnGα,δ(t,x)=t-n+1/αGα,δ(1,t-1/αx).(3)
Gα(s,·)∗Gα,δ(t,·)=Gα,δ(s+t,·) for any s,t∈(0,∞).(4)
Forn≥1, there exist some constants C and Cn>0 such that, for all x∈R, (29)Gα,δ1,x≤C11+x1+α,∂n∂xnGα,δ1,x≤Cnxα+n-11+xα+n2.(5)
∫0T‍∫R‍|Gα,δt,x|λdtdx<∞ if and only if 1/α<λ<α.Ford≥1 and any multi-index α=(α1,…,αd) and δ=(δ1,…,δd), denote by Gα,δ(t,x) the Green function of the deterministic equation as follows:(30)∂u∂tt,x=Dδαt,x,u0,x=δ0x,t>0,x∈Rd.Clearly, (31)Gα,δt,x=∏i=1d‍Gαi,δi(t,xi)=12πd∫Rd‍exp⁡⁡t∑i=1d‍-iξ,x-t∑i=1d‍ξiαihhhhhhhhhhhhhh×exp⁡-iδπ2sgnξit∑i=1d‍dξ.Now, we describe a solution of (2) in Sρ sense.Definition 4.
We say that a stochastic field{u(t,x),(t,x)∈[0,T]×Rd} is a solution of (2) in Sρ sense, if (1)
u:R+×R×Ω is jointly measurable;(2)
∫0t‍∫Rd‍Gα,δ(t-s,x-z)u(s,z)dBH(s,z) is well defined for all t∈R+ and x∈Rd as an element of Sρ for certain ρ∈R;(3)
the following equation holds inSρ: (32)ut,x=Gα,δ(t)∗u(0,x)+∫0t∫Rd‍Gα,δ(t-s,x-z)u(s,z)dBH(s,z).The following embedding proposition given by Mémin et al. [45] is useful for our derivations below.Lemma 5.
Ifh∈(1/2,1) and f,g∈L1/h([a,b]), one has (33)∫ab‍∫ab‍fugvu-v2h-2dudv≤ChfL1/h([a,b])gL1/h([a,b]),where Ch>0 is a constant depending only on h.


## 2.3. Malliavin Calculus

In this subsection, let us brief the basic elements of the Malliavin calculus (see, e.g., Nualart [29]).SinceBH(t,x), (t,x)∈[0,T]×Rd, is Gaussian, we might develop the Malliavin calculus with respect to fractional noises in order to prove the existence of the laws of the solutions of SPDE driven by fractional noises.LetBH(φ)=∫0T‍∫Rd‍φ(t,x)BH(dt,dx) for φ∈H1 and (34)S=F=fBHφ1,…,BHφn;hf∈Cb∞(Rn),φi∈H1,n≥1F=fBH(φ1),…,BH(φn);be the space of all “smooth cylindrical” random variables, where Cb∞(Rn) denotes the class of all bounded infinitely differentiable functions on Rn, whose partial derivatives are also bounded. The Malliavin derivative of an element F=f(BHφ1,…,BHφn), with respect to BH, is defined by (35)Dt,xF≔∑i=1n‍∂f∂xiBHφ1,…,BHφnφi(t,x).Let D1,2 be the completion of S under the norm (36)F1,22≔EF2+EDFH12.Then, D1,2 is the domain of the closed operator D. For each h∈H1 and F∈S, define (37)DhF≔limɛ→0ddhfBH(h1)+ɛh1,hH1,…,BH(hn)hhhhhhh+ɛhn,hH1,which might be extended as a closed operator on L2(Ω) with the domain Dh being the closure of S under the norm(38)Fh2=E[F2+DhF2].Let {hn,n≥1} be an orthonormal basis of H1. Then, F∈D1,2 if and only if F∈Dhn for each n∈N and(39)∑n=1∞‍EDhnF2<∞.In this case, (40)DhF=DF,hH1.On the other hand, the divergence operatorδ:Domδ⊂L2(Ω;H1)→L2(Ω) is the adjoint of the operator D and is uniquely defined by the following duality relationship: u∈Domδ if and only if(41)E(Fδ(u))=EDF,uH1,∀F∈S.Note that u∈Domδ if and only if u is integrable with respect to BH. In the literature, δ is called the Skorokhod integral with respect to BH.The following propositions (see Wei [46] for the case of fractional noises), which are the Malliavin calculus with respect to fractional-colored noises, can deduce the laws for solutions to the corresponding stochastic partial differential equations.Proposition 6.
LetA∈F. If F is a square integrable random variable that is measurable with respect to the σ-field FAc, then (42)DF1A=0,a.s.Remark 7.
Let{u(t,x),(t,x)∈[0,T]×Rd} be an {Ft,t∈[0,T]}-adapted random field. By Proposition 6, we have Ds,yu(t,x)=0, a.s., for any 0≤t<s≤T and x,y∈Rd.Proposition 8.
SupposeF∈D1,2; if DFH12>0, a.s., then the distribution of the random variable F is absolutely continuous with respect to Lebesgue measure.Remark 9.
Propositions6 and 8 can be proved similarly to those in Nualart [29].


## 3. Lyapunov Exponent Estimate of the Solution

In this section, we will establish the existence and uniqueness of the solution of (2) and give the Lyapunov exponent estimate of the solution. We denote Tn={0<s1<s2<⋯<sn<t} and ds=ds1ds2⋯dsn. In fact, iterating (32) yields formally that(43)u(t,x)=Gα,δ(t)∗u(0,x)+∑n=1∞‍In(fn(t,x)),where(44)fn(t,x;s1,x1,s2,x2,…,sn,xn)Sym∫Rd‍Gα,δ(t-sn,x-xn)⋯Gα,δ(s2-s1,x2-x1)Symhhhh×Gα,δ(s1,x1-y)u(0,y)dy∫Rd‍,where Sym is the symmetrization with respect to n ((d+1)-dimensional) variables (s1,x1),…,(sn,xn) and(45)Infnt,x=∫0,T×Rdn‍fn(t,x;s1,x1,s2,x2,…,sn,xn)H·dBH(s1,x1)dBH(s2,x2)⋯dBH(sn,xn).We compute the L2 norm of each chaos; that is, (46)Θn(t,x)=EInfnt,x2.Assume that |u(0,x)|≤C<∞, where, in what follows, inessential constants will be denoted generically by C, even if they vary from line to line. By the isometric equality (18), we obtain (47)Θnt,x=n!∫Tn2‍∫R2nd+1‍∏i=1nφh0si-riφHξi-ηiHHHH×Gα,δt-sn,x-ξnHHHH⋯Gα,δs2-s1,ξ2-ξ1HHHH×Gα,δ(s1,ξ1-y)u0(y)HHHH×Gα,δ(t-rn,x-ηn)HHHH⋯Gα,δ(r2-r1,η2-η1)HHHH×Gα,δ(r1,η1-y)HHHH×u0y~dydy~dξdηdsdr≤Cn!∫Tn2‍∫R2nd‍∏i=1nφh0(si-ri)φH(ξi-ηi)hhhhhh×Gα,δ(t-sn,x-ξn)hhhhhh⋯Gα,δ(s2-s1,ξ2-ξ1)hhhhhh×Gα,δ(t-rn,x-ηn)⋯Gα,δhhhhhh×r2-r1,η2-η1dξdηdsdr.Let (48)Gαi,δit,x=12π∫R‍exp⁡-iδiπ2sgnz-izx-tzαihhhhhhhhhhi×exp⁡-iδiπ2sgnzdz,hhhhhhhhhhhhhhhhhhhhhhhi1≤i≤d.Thus, (49)Θn(t,x)=Cn!∫Tn2‍∏i=1nφh0si-ri∏i=1dΘi,nt,xi,s,rdsdr,where (50)Θi,n(t,xi,s,r)=∫R2n‍∏k=1nφhi(ρk-τk)Gαi,δi(t-sn,xi-ρn)⋯Gαi,δi(s2-s1,ρ2-ρ1)·Gαi,δi(t-rn,xi-τn)⋯Gαi,δi(r2-r1,τ2-τ1)hhhhhhh×dρ1dρ2⋯dρndτ1dτ2⋯dτn.Now, we estimate Θi,n(t,xi,s,r).Lemma 10.
There is a constantChi such that(51)Θi,n(t,xi,s,r)≤Chin∏k=1n‍sk+1-skhi-1/αirk+1-rkhi-1/αi,where sn+1=rn+1=t.Proof.
Applying Lemma5, we have (52)Θ~i,nt,xi,s1,r1=∫R2‍φhiρ1-τ1Gαi,δis2-s1,ρ2-ρ1hhhh×Gαi,δi(r2-r1,τ2-τ1)dρ1dτ1=hi(2hi-1)∫R2ρ1-τ12hi-2×Gαi,δi(s2-s1,ρ2-ρ1)×Gαi,δi(r2-r1,τ2-τ1)dρ1dτ1≤CGαi,δis2-s1,ρ2-·L1/hi(R)×Gαi,δir2-r1,τ2-·L1/hi(R).Note that(53)Gαi,δis2-s1,ρ2-·L1/hi(R)=∫R‍Gαi,δis2-s1,ρ2-ρ11/hidρ1hi=∫R‍s2-s1-1/hiαiGαi,δi1,s2-s1-1/αiρ2-ρ11/hihhhhh×dρ1∫R‍hi=∫R‍s2-s11/αi-1/hiαiGαi,δi1,ρ11/hidρ1hi≤s2-s11/αi(hi-1)∫R‍c1+ρ121/hidρ1≤Chis2-s11/αihi-1.Therefore, (54)Θ~i,n(t,xi,s1,r1)≤Chi,αi,δis2-s1r2-r11/αi(hi-1).Thus, we show that (51) is true when n=1. The lemma follows from iteration.Lemma 11.
Let2h0-∑i=1d‍1-hi/αi>1 and p∈(1/2h0-1,1/∑i=1d‍1-hi/αi). Then, u(t,x)∈Sρ if ρ<2/p-2∑i=1d‍1-hi/αi-1. Moreover,(55)limsup⁡t→∞log⁡sup⁡x∈Rd⁡u(t,x)ρ2tk<∞,for k=2(h0-∑i=1d‍(1-hi)/αi)/(2/p-2∑i=1d‍(1-hi)/αi-1-ρ).Proof.
Letγ=2(∑i=1d‍(1-hi)/αi), p, q>1, and 1/p+1/q=1. Then, by the Hölder inequality, one gets(56)Θn(t,x)≤n!CH,α,δn∫Tn2‍∏k=1n‍sk-rk2h0-2hhhhhhhhhhh×sk+1-skrk+1-rk-γ/2dsdr≤n!CH,α,δn∫Tn2‍∏k=1nsk+1-skrk+1-rk‍-pγ/2dsdr1/p·∫Tn2‍∏k=1n‍sk-rk2h0-2qdsdr1/q=n!CH,α,δn∫Tn‍∏k=1n‍sk+1-sk-pγ/2ds2/p·∫Tn2∏k=1n‍sk-rk2h0-2qdsdr1/q.Following Hu [30], one can get a constant Cp,h0 such that, for any value q such that 1<q<1/2-2h0 (i.e, p>1/2h0-1),(57)∫Tn‍∏k=1n‍sk-rk2h0-2qdsdr1/q≤Cp,h0nt(2+(2h0-2)q)n/q.Set θ=pγ/2∈(0,1) (i.e, p<1/∑i=1d‍1-hi/αi); similar to the proof of (3.6) in Bo et al. [37], we get (58)∫Tn‍∏k=1n‍sk+1-sk-θds≤CHnt(1-θ)nΓ(n(1-θ)+1),where Γ(x)=∫0∞‍e-ttx-1 is the Gamma function. Thus, we obtain (59)Θn(t,x)≤CH,α,δnn!t2n/p1-p∑i=1d1-hi/αiΓn1-p∑i=1d1-hi/αi+12/p·t(2+(2h0-2)q)n/q=CH,α,δnn!t2n(h0-∑i=1d1-hi/αi)Γn1-p∑i=1d1-hi/αi+12/p.We continue to use the notation introduced previously. The Stirling formula yields that, for x>-1, (60)Γ(x+1)=K(x)xxe-x,where the function K(x) satisfies λ-x≤K(x)≤λx. Hence, for β=1-p∑i=1d‍1-hi/αi, by the definition of Sρ, (61)ut,xρ2≤∑n=0∞n!ρΘn(t,x)≤∑n=0∞n!ρn!CH,α,δnt2nh0-∑i=1d1-hi/αiΓnβ+12/p≤∑n=0∞CH,α,δnλ4β/pn2β/p-ρ+12β/p-ρ+1β2β/pn×t2nh0-∑i=1d1-hi/αiΓk2β/p-ρ+1+1≤E2β/p-ρ+1C~t2h0-∑i=1d1-hi/αi,where C~=CH,α,δλ4β/p(2β/p-(ρ+1))2β/p-(ρ+1)/β2β/p>0 and Er(z) is the Mittag-Leffler function with the parameter r>0. Note that 2β/p-ρ-1>0; then, by the asymptotic property of the Mittag-Leffler function (see, e.g., Podlubny [1]), one has(62)ut,xρ2≤C2β/p-ρ+1×exp⁡(C~t2(h0-∑i=1d1-hi/αi)/2β/p-(ρ+1))<∞,if ρ<2/p-2∑i=1d1-hi/αi-1. This completes the proof of the lemma.Lemma 12.
u
(
t
,
x
) defined by (43) is the solution of (2) in the sense of Definition 4.Proof.
Letu(t,x) be given by (43). It suffices to verify (32). By the Fubini lemma and the definition of the integral and the definition of fn(t,x), we have (63)∫0t‍∫RdGα,δ(t-s,x-z)In(fn(s,z))dBH(s,z)‍=In+1(fn+1(t,x)).Thus, the Lemma follows.From the above lemmas, we get the following.Theorem 13 (existence, uniqueness, and Lyapunov exponent estimate).
Let2h0-∑i=1d‍1-hi/αi>1 and p∈(1/2h0-1,1/∑i=1d‍1-hi/αi). If u0∈L∞(Rd), then (2) has a unique solution u(t,x)∈Sρ when ρ<2/p-2∑i=1d1-hi/αi-1. Moreover,(64)limsup⁡t→∞log⁡sup⁡x∈Rd⁡u(t,x)ρ2tk<∞,for k=2n(h0-∑i=1d‍1-hi/αi)/(2/p-2∑i=1d‍1-hi/αi-1-ρ).


## 4. Hölder Regularity

Theorem 14.
Assume that2h0-∑i=1d‍1-hi/αi>1. If there exists some γ∈(0,1) such that, for all z∈K (compact subset of Rd), (65)sup⁡x∈RdE(u0x+z-u0xP)<czγp0for some p0 large enough in [2,α0/γ), where α0=min⁡1≤i≤d{αi}, then the solution {u(t,x),(t,x)∈[0,T]×Rd} is θ Hölder continuous in t and μ Hölder continuous in x, where θ∈(0,min⁡1≤i≤d{1,∑j=1d‍γ/αj,h0αi(1+∑j=1d‍((hj-1)/h0αj))}) and μ∈(0,min⁡{γ,(h0-∑i=1d‍(1-hi/αi))(1+∑i=1d‍1/αi)-1}).Proof.
It follows from the proof of Lemma11. If p∈(1/2h0-1,2(1+2∑i=1d‍1-hi/αi)-1), then u(t,x)∈Sρ when ρ<2/p-2∑i=1d‍1-hi/αi-1. In particular, u(t,x)∈S0=L2(Ω) since 2/p-2∑i=1d1-hi/αi-1>0. On the other hand, from (43), it follows that (66)ut,x=Gα,δ(t)∗u(0,x)+∑n=1∞‍In(fn(t,x))=Gα,δ(t)∗u(0,x)+∫0t‍∫Rd‍Gα,δ(t-r,x-z)u(s,z)dBH(r,z).Thus, for s,t∈[0,T] and x,y∈Rd, we have(67)Eut,x-us,y2≤EGα,δt∗u0,x-Gα,δs∗u0,y2+E∫st‍∫Rd‍Gα,δt-r,x-zur,zdBHr,z2+E∫0s‍∫Rd‍Gα,δt-r,x-z-Gα,δs-r,y-zhhhhhhhhh×ur,zdBHr,z∫Rd‍2≔I+II+III.Now, we will estimate A, B, and C, respectively. Following Boulanba et al. [20], one gets(68)I≤CTx-y2γ1+t-s2γ2,where 0<γ1<ρ and 0<γ2<min⁡{∑i=1d‍γ/αi,1}.
By Lemma5 and (62) with ρ=0, one can get (69)II=∫[s,t]2‍∫R2d‍φh0r-r¯φHz-z¯Gα,δt-r,x-zuUUUU×Gα,δt-r¯,x-z¯uUUUU×Eur,zur¯,z¯dzdz¯drdr¯≤∫[s,t]2‍∫R2d‍φh0r-r¯φHz-z¯Gα,δt-r,x-zhhhhhhhh×Gα,δt-r¯,x-z¯ur,z0hhhhhhhh×ur¯,z¯0dzdz¯drdr¯≤CT∫[s,t]2‍∫R2d‍φh0r-r¯φHz-z¯Gα,δt-r,x-zhhhhhhhhh×Gα,δt-r¯,x-z¯dzdz¯drdr¯=CT∫[s,t]2‍φh0r-r¯drdr¯×∫R2d‍φHz-z¯Gα,δt-r,x-zhhhhh×Gα,δt-r¯,x-z¯dzdz¯≤∑uth22∫st‍∑∫R‍∫R‍⋯∫R‍Gα,δt-r,x-z1/hddzdhd/hd-1hhhhhhhhh⋯dz2∫R‍Gα,δt-r,x-z1/hddzdhd/hd-1h2/h1dz1∑h1/h0dr2h0≕CTGα,δ(t-·,x-·)L1/hd⋯1/h11/h02.By the definition of the Green function Gα,δ and Lemma 3, one gets (70)∫R‍Gα,δt-r,x-z1/hddzd=∏i=1d-1‍Gαi,δit-r,xi-zi1/hd×∫R‍t-r-1/αdhd×Gαi,δi1,t-r-1/αdhdxi-zi1/hddzd=∏i=1d-1‍Gαi,δit-r,xi-zi1/hd×∫R‍t-r-1/αdhd+1/αdGαi,δi1,zd1/hddzd≤∏i=1d-1‍Gαi,δit-r,xi-zi1/hdhhhhhh×t-r-1/αdhd+1/αd×∫R‍C1+zd21/hddzd≤C∏i=1d-1‍Gαi,δit-r,xi-zi1/hdt-r-1/αdhd+1/αd.Using induction, one can get (71)Gα,δt-·,x-·L1/hd⋯1/h11/h02=∑uth22∫st‍∑444∫R‍∫R‍⋯∫R‍Gα,δt-r,x-z1/hddzdhd/hd-1hhhhhhhhhhhhhhhh⋯dz2∫R‍Gα,δt-r,x-z1/hddzdhd/hd-1h2/h1dz1∑444h1/h0dr2h0≤CT,H,α,δ∫st‍t-r∑i=1dhi-1/h0αidr2h0≤CT,H,α,δt-s2h0∑i=1dhi-1/h0αi+1.Therefore,(72)II≤CT,H,α,δt-s2h0∑i=1dhi-1/h0αi+1.As for III, we have(73)III≤2E∫0s‍∫Rd‍Gα,δt-r,x-zhhhhhhhhhhhh-Gα,δt-r,y-zhhhhhhhh×ur,zdBHr,z∫Rd‍2hh+E∫0s‍∫Rd‍∫Rd‍Gα,δt-r,y-zhhhhhhhhhhhhh-Gα,δs-r,y-z∫Rd‍hhhhhhhhhhhh×ur,zdBHr,z∫Rd‍2≕III1+III2.Let θ∈(0,1). Then,(74)III1=∫[0,s]2‍∫R2d‍φh0(r-r¯)φH(z-z¯)hhhhhh×Gα,δt-r,x-z-Gα,δt-r,y-zhhhhhh×Gα,δt-r¯,x-z¯-Gα,δt-r¯,y-z¯hhhhhh×Eur,zur¯,z¯dzdz¯drdr¯≤CT∫[0,T]2‍∫R2d‍φh0(r-r¯)φH(z-z¯)i×Gα,δt-r,x-zhhhhhhhhhhhh-Gα,δt-r,y-zhhhhhhhh×Gα,δ(t-r¯,x-z¯)hhhhhhhhhhh-Gα,δt-r¯,y-z¯dzdz¯drdr¯=CT∫[0,T]2‍φh0r-r¯drdr¯×∫R2d‍φH(z-z¯)hhhh×Gα,δt-r,x-z-Gα,δt-r,y-zhhhh×Gα,δt-r¯,x-z¯hhhhhhhhh-Gα,δt-r¯,y-z¯dzdz¯=CTGα,δt-·,x-·-Gα,δt-·,y-·L1/hd⋯1/h11/h02≤CT∫rGα,δt-·,x-·-Gα,δt-·,y-·θhhhhhh×Gα,δt-·,x-·1-θL1/hd⋯1/h11/h02hhhhi+Gα,δt-·,x-·-Gα,δt-·,y-·θhhhhhhhhh×Gα,δt-·,y-·1-θL1/hd⋯1/h11/h02∫r≔CT(III1,1+III1,2).Using the mean-value theorem, it holds that(75)III1,1≤x-yθ∇Gα,δt-·,ξ-·θhhh×Gα,δt-·,x-·1-θL1/hd⋯1/h11/h02=x-y2θ∑uth22∑i=1d∂∂xiGα,δt-·,ξ-·θhhhhhhhhhh×Gα,δt-·,x-·1-θ∑uth22L1/hd⋯1/h11/h02≤x-y2θ∑i=1d‍∂∂xiGα,δt-·,ξ-·θ2∑i=1d‍∂∂xiGα,δt-·,ξ-·θhhhhhhhhhhi×Gα,δt-·,x-·1-θ∑i=1d‍∂∂xiGα,δt-·,ξ-·θL1/hd⋯1/h11/h02≤CT,H,θx-y2θ×∑i=1d∂∂xiGα,δt-·,ξ-·θhhhhhhhhh×Gα,δt-·,x-·1-θ∂∂xiGα,δt-·,ξ-·θL1/hd⋯1/h11/h02.Using Lemma 3, one has(76)∫R‍∂∂xdGα,δt-r,ξ-zθ/hdhhh×Gα,δt-r,x-z1-θ/hddzd=∏i=1d-1‍Gαi,δit-r,ξi-zi1/hd×∫R‍t-r-2θ/αdhd-1-θ/αdhdhhhh×∂∂xdGαd,δd1,t-r-1/αdξd-zdθ/hdhhhh×Gαd,δd1,t-r-1/αdξd-zd1-θ/hddzd=∏i=1d-1‍Gαi,δit-r,xi-zi1/hd×∫R‍t-r-1-θ/αdhd+1/αdhhhhh×∂∂xdGαd,δd1,ξd-zdθ/hdiiiiiiiii×Gαd,δd1,zd1-θ/hddzd=∏i=1d-1‍Gαi,δit-r,xi-zi1/hdhhhhhh×t-r-2θ/αdhd-1-θ/αdhd+1/αd×∫R‍∂∂xdGαd,δd1,ξd-zdθ/hdhhhhh×Gαd,δd1,zd1-θ/hddzd≤C∏i=1d-1‍Gαi,δit-r,xi-zi1/hd×t-r-2θ/αdhd-1-θ/αdhd+1/αd.Using induction, one can get(77)∑uth22∂∂xiGα,δt-·,y-·θ×Gα,δt-·,x-·1-θ∑uth22L1/hd⋯1/h11/h02=∑uth22∫0T‍∑uth∫R‍∫R‍⋯∫R‍∂∂xiGα,δt-r,y-zθ/hd×Gα,δt-r,x-z1-θ/hddzd∫R‍hd/hd-1⋯dz2∫R‍h2/h1dz1∑uthh1/h0dr2h0≤CT,H,α,δ∫0T‍t-r∑j=1dhj-1/h0αj-θ/αih0dr2h0.Therefore, if ∑j=1d‍hj-1/h0αj-θ/αih0+1>0, that is, θ<h0αi(1+∑j=1d‍hj-1/h0αj), i=1,2,…,d, we have(78)III1,1≤CT,H,θ,α,δx-y2θ.Similarly,(79)III1,2≤CT,H,θ,α,δx-y2θ.Together with (78)-(79), one gets(80)III1≤CT,H,θ,α,δx-y2θ.As for III2, by letting μ∈(0,1), we have(81)III2=E∫0s‍∫Rd‍Gα,δt-r,y-z-Gα,δs-r,y-zhhhhhhh×ur,zdBHr,z∫Rd‍2=∫[0,s]2‍∫R2d‍φh0(r-r¯)φH(z-z¯)hhhhhh×Gα,δt-r,y-z-Gα,δs-r,y-zhhhhhh×Gα,δt-r¯,y-z¯-Gα,δs-r¯,y-z¯hhhhhh×Eur,zur¯,z¯dzdz¯drdr¯≤CT∫[0,T]2‍φh0r-r¯drdr¯×∫R2d‍φHz-z¯hhhh×Gα,δt-r,y-z-Gα,δs-r,y-zhhhh×Gα,δ(t-r¯,y-z¯)hhhhhhhh-Gα,δs-r¯,y-z¯dzdz¯=CTGα,δt-·,y-·hhhhh-Gα,δs-·,y-·L1/hd⋯1/h11/h02≤CTGα,δs-·,y-·1-μL1/hd⋯1/h11/h02Gα,δt-·,y-·-Gα,δs-·,y-·μhhhhh×Gα,δt-·,y-·1-μL1/hd⋯1/h11/h02hhhh+Gα,δt-·,y-·-Gα,δs-·,y-·μhhhhhhhh×Gα,δs-·,y-·1-μL1/hd⋯1/h11/h02≔CTIII2,1+III2,2.By the mean-value theorem, it holds that (82)III2,1=t-sμ∂∂tGα,δρ-·,y-·μhh·Gα,δt-·,y-·1-μ∂∂tGα,δρ-·,y-·μL1/hd⋯1/h11/h02.Note that (83)Gα,δ(t,y)=t-∑i=1d1/αiGα,δ(1,t-∑i=1d1/αiy).Hence, one can attain (84)∂∂tGα,δt,y=-∑i=1d‍1αit-1-∑i=1d1/αiGα,δ(1,t-∑i=1d1/αiy)+-∑i=1d‍1αit-1-2∑i=1d1/αi×∑i=1d‍yi∂∂yiGα,δ1,t-∑i=1d1/αiy.After induction, therefore, one has (85)∂∂tGα,δρ-·,y-·μ·Gα,δt-·,y-·1-μL1/hd⋯1/h11/h02≤CT∑i=1d‍2∂∂tGα,δρ-·,y-·μt-r-1-∑i=1d1/αihhhhhhhhh×Gα,δ1,ρ-r-∑i=1d1/αiy-z∂∂tGα,δρ-·,y-·μμhhhhhhhh·Gα,δt-r,y-z1-μ∂∂tGα,δρ-·,y-·μL1/hd⋯1/h11/h02+∑i=1d‍t-r-1-2∑i=1d1/αihhhhh×∑i=1d‍yi-zi∂∂yihhhhhhhhhh×Gα,δ1,ρ-r-∑i=1d1/αiy-z∑i=1d‍μhhhhhh·Gα,δt-r,y-z1-μ∑i=1d‍L1/hd⋯1/h11/h02=CT∑333222∫0T‍∑555∫R‍∫R‍hhhhhhhhhhhhi⋯∫R‍t-r-1-∑i=1d1/αihhhhhhhhhhhiiiihiii×Gα,δ1,ρ-r-∑i=1d1/αiy-zμ/hdhhhhhhhhhhhiiiihiii·Gα,δt-r,y-z1-μ/hddzd∫R‍hd/hd-1hhhhhhhhhhhhi⋯dz2∫R‍h2/h1dz1∑555h1/h0dr2h0+CT∑333222∫0T‍∫R‍∫R‍∑i=1dyi-zihhhhhhhhhhhhhhhii⋯∫R‍∑i=1dyi-zit-r-1-2-∑i=1d1/αihhhhhhhffdffhhhhhh×∑i=1dyi-zi∂∂yiGα,δhhhhhhhhhhhhhhhhh×1,ρ-r-∑i=1d1/αiy-z∑i=1dyi-ziμ/hdhhhhhhhhhhhhh×·Gα,δt-r,y-z1-μ/hddzd∫R‍hd/hd-1hhhhhhhh⋯dz2∑i=1dyi-zih2/h1dz1∑555h1/h0dr2h0≤CT,H,α,δ∫0T‍t-r-μ-∑i=1d1-hi/αi1/h0dr2h0+CT,H,α,δ∫0Tt-r-μ1+∑i=1d1/αi-∑i=1d1-hi/αi1/h0dr2h0.Thus, if(86)-μ1+∑i=1d‍1αi-∑i=1d1-hiαi1h0+1>0,that is,(87)μ<h0-∑i=1d1-hi/αi1+∑i=1d1/αi,then(88)III2,1≤Cα,δ,H,T,μt-s2μ.Similarly, we can check(89)III2,2≤Cα,δ,H,T,μt-s2μ.From the estimates (78)-(79), it follows that(90)III2≤Cα,δ,H,T,μt-s2μ.This completes the proof of the theorem.


## 5. Absolute Continuity of the Solution

In this section, we will prove that the law of the solution for (2) is absolutely continuous with respect to Lebesgue measure on Rd.Theorem 15.
Assume that2h0+∑i=1d‍hi-1/αi>1 and ∑i=1d‍1/αi<1. Then, for (t,x)∈[0,T]×Rd, the law of the solution u(t,x) of (2) is absolutely continuous with respect to the Lebesgue measure on Rd.Let us start with two lemmas for the proof of our main result.Lemma 16.
Assume2h0+∑i=1d‍hi-1/αi>1. For (t,x)∈[0.T]×Rd, the solution u(t,x)∈D1,2. Its Malliavin derivative satisfies, if s≤t,(91)Ds,yut,x=Gα,δ(t-s,x-y)u(s,y)+∫st‍∫Rd‍Gα,δ(t-r,x-z)Ds,yu(r,z)dBH×(r,z).And if s>t, Ds,yu(t,x)=0.Proof.
For eachn≥1, let {un(t,x),n≥1} be the Picard iteration defined as(92)u0(t,x)=∫Rd‍Gα,δ(t,x-y)u(0,y)dy,(93)un+1t,x=u0(t,x)+∫0t‍∫Rd‍Gα,δ(t-s,x-y)un(s,y)dBH×(s,y).Now, we will prove un(t,x)∈D1,2 by induction. Suppose that, for n≥0 and (t,x)∈[0,T]×Rd, un(t,x)∈D1,2 and satisfies(94)sup⁡t,x∈0,T×RdE∫0T‍∫Rd‍Ds,yunt,x2dyds<∞.We take the Malliavin derivative at both sides of (93). Then, for s≤t,(95)Ds,yun+1t,x=Gα,δ(t-s,x-y)un(s,y)+∫st‍∫Rd‍Gα,δ(t-r,x-z)Ds,yhhhhh×un(r,z)dBH(r,z)and, for s>t, Ds,yun+1(t,x)=0. Note that(96)E∫0t‍∫Rd‍Ds,yun+1t,x2dyds=E∫0t‍∫Rd‍∫0t‍Gα,δt-s,x-yuns,yhhhhhhhhhhh+∫st‍∫Rd‍Gα,δ(t-r,x-z)hhhhhhhhhhhhhh×Ds,yunr,zdBHr,z∫0t‍2dyds∫0t‍≤2Ds,yunr,zdBHr,z∫st‍2E∫0t‍∫Rd‍Gα,δt-s,x-yuns,y2dydshh+E∫0t‍∫Rd‍∫st‍∫Rd‍Gα,δ(t-r,x-z)×Ds,yunr,zdBHr,z∫st‍2dyds:=2(Bn1(t,x)+Bn2(t,x)).Recall that un(t,x)∈S0; that is, un(t,x)∈L2(Ω). Thus, we have(97)Bn1(t,x)=∫0t‍∫Rd‍EGα,δt-s,x-yuns,y2dyds≤∫0T‍∫Rd‍Gα,δt-s,x-y2dyds<∞.By the Fubini theorem, one gets(98)Bn2t,x=Eunr,zdBHr,z∫0t‍2∫0t‍∫Rd‍∫st‍∫Rd‍Gα,δt-r,x-zDs,yhhhhhhhhhhhhhi×unr,zdBHr,z∫0t‍2dyds=∫0t‍∫Rd∫[s,t]2‍∫R2d‍φh0(r-r¯)φH(z-z¯)‍hhhhhhhhhhhhhhh×Gα,δt-r,x-zhhhhhhhhhhhhhhh×Gα,δt-r¯,x-z¯hhhhhhhhhhhhhhh×EDs,yunr,zDs,yunr¯,z¯hhhhhhhhhhhhhhh×dzdz¯drdr¯∫[s,t]2‍dyds≤C∫0t‍∫Rd‍∫[s,t]2‍∫R2d‍φh0r-r¯φHz-z¯hhhhhhhhhhhhhhh×Gα,δt-r,x-zhhhhhhhhhhhhhhh×Gα,δt-r¯,x-z¯hhhhhhhhhhhhhhh×EDs,yunr,z2dzdz¯drdr¯∫[s,t]2‍dyds≤C∫[s,t]2‍∫R2dsup⁡z∈Rd∫0t‍∫Rd‍EDs,yunr,z2dyds‍hhhhhhhhhhh×φh0r-r¯φHz-z¯hhhhhhhhhhh×Gα,δt-r,x-zhhhhhhhhhhh×Gα,δt-r¯,x-z¯dzdz¯drdr¯.Note that (99)∫R2d‍φHz-z¯Gα,δt-r,y-zhhh×Gα,δ(t-r¯,y-z¯)dzdz¯=∫R2d-1‍∏i=1d-1‍φhizi-z¯iGαi,δit-r,xi-zihhh×Gαi,δit-r¯,xi-z¯idz1dz¯1⋯dzd-1dz¯d-1hhh×∫R2‍φhd(zd-z¯d)Gαd,δd(t-r,xd-zd)hhh×Gαd,δd(t-r¯,xd-z¯d)dzddz¯d,∫R2‍φhd(zd-z¯d)Gαd,δd(t-r,xd-zd)×Gαd,δd(t-r¯,xd-z¯d)dzddz¯d≤CGαd,δd(t-r,xd-·)L1/hd×Gαd,δdt-r¯,xd-·L1/hd=C(t-r)hd-1/αd(t-r¯)hd-1/αd.So by induction, if 2h0+∑i=1d‍hi-1/αi>1, we have(100)Bn2(t,x)≤C∫[s,t]2‍sup⁡z∈Rd∫0t‍∫Rd‍EDs,yunr,z2dydshhhhhh×φh0r-r¯×t-r∑i=1dhi-1/αihhhhhh×t-r¯∑i=1dhi-1/αidrdr¯≤C∫0t‍sup⁡z∈Rd(∫0t‍∫Rd‍EDs,yunr,z2dyds)hh×t-r∑i=1dhi-1/αihhih×(∫0r‍r-r¯2h0-2t-r¯∑i=1dhi-1/αidr¯)dr≤C∫0t‍sup⁡z∈Rd∫0t‍∫Rd‍EDs,yunr,z2dydshhhhhh×t-r∑i=1dhi-1/αidr=C∫0t‍sup⁡z∈Rd(∫0t‍∫Rd‍EDs,yunr,z2dyds)hhhhhh×g1(t-r)dr,where ∫0T‍t∑i=1d‍hi-1/αidt<+∞, since ∑i=1d‍hi-1/αi+1>1+(1-2h0)>0. Together with (96)–(100), one gets (101)sup⁡z∈RdE∫0t‍∫Rd‍Ds,yun+1t,x2dyds≤C+C∫0t‍sup⁡z∈Rd(∫0t‍∫Rd‍EDs,yunr,z2dyds)×g1(t-r)dr.By the version of Gronwall’s lemma given by Dalang [47], one has(102)sup⁡n≥0sup⁡t,x∈0,T×RdE∫0T‍∫Rd‍Ds,yunt,x2dyds<∞.Note that EDun(t,x)H1>0, if and only if EDun(t,x)L2([0,T]×Rd)>0. Equation (102) implies EDun(t,x)H1>0. On the other hand, we note that un(t,x)∈S0; that is, un(t,x)∈L2(Ω). Thus, we have proved un(t,x)∈D1,2. By Lemma 1.2.3 in Nualart [29], we get u(t,x)∈D1,2 and the sequence Dun(t,x) converges to Du(t,x) in the weak topology of L2([0,T]×Rd). This concludes the proof of Lemma 16.Lemma 17.
Let∑i=1d‍1/αi<1. Then, there exists some constant C>0 such that, for 0<ɛ<t,(103)sup⁡s,y∈t-ɛ,t×RdE∫t-ɛt‍∫Rd‍Dr,zus,y2dzdr<Cɛ1-∑i=1d1/αi.Proof.
Fors∈[t-ɛ,t], set Iɛ(s,y)=E(∫t-ɛs‍∫Rd‍|Dr,zus,y|2dzdr). By (102), we have (104)sup⁡(s,y)∈[0,T]×RdIɛ(s,y)<∞.Recall (91). Then,(105)Iɛs,y≤2E∫t-ɛs‍∫Rd‍Gα,δs-r,y-zus,y2dzdrhhhh+E∫t-ɛs‍∫Rd‍∫rs‍∫Rd‍Gα,δs-r1,y-x1hhhhhhhhhhhhhhhhhhh×Dr,zur1,x1dBHhhhhhhhhhhhhhhhhhhh×r1,z1∫t-ɛs‍2dzdr:≡2(Iɛ,1(s,y)+Iɛ,2(s,y)).Note that(106)Iɛ,1s,y=∫t-ɛs‍∫Rd‍EGα,δs-r,y-zus,y2dzdr≤∫t-ɛt‍∫Rd‍Gα,δs-r,y-z2dzdr≤C∫t-ɛt‍s-r-∑i=1d1/αidr<Cɛ1-∑i=1d1/αi,(107)Iɛ,2(s,y)≤Cɛ1-∑i=1dhi-1/αi∫t-ɛs‍sup⁡x1∈RdIɛ(r1,x1)dr1.Together with (105)–(107), one gets(108)sup⁡y∈RdIɛs,y≤ɛ1-∑i=1d1/αi+Cɛ1-∑i=1dhi-1/αi×∫t-ɛs‍sup⁡x1∈RdIɛ(r1,x1)dr1.Then by the Gronwall’s lemma, we get (103).Proof of Theorem15.
In order to prove Theorem15, by Theorem 2.1.3 in Nualart [29], we only need to check that (109)Du(t,x)H1>0,a.s.Note that Du(t,x)H1>0, a.s., if and only if Du(t,x)L2([0,T]×Rd)>0. Hence, we only need to prove that(110)∫0t‍∫Rd‍Dr,zut,x2dzdr>0,a.s.In (91), we note that(111)∫0t‍∫Rd‍Dr,zut,x2dzdr≥∫t-ɛt‍∫Rd‍Dr,zut,x2dzdr≥CJɛ,1t,x-Jɛ,2t,x,where(112)Jɛ,1t,x=∫t-ɛt‍∫Rd‍Gα,δt-r,x-zur,z2dzdr,Jɛ,2t,x≔∫t-ɛt‍∫Rd∫rt‍∫Rd‍Gα,δt-r1,x-z1‍hhhhhhhhhhhhh×Dr,zur1,z1dBHr1,z1∫rt‍∫Rd‍2dzdr.Similar to the proof of (106), there exists some constant C>0 such that(113)EJɛ,1t,x≤C∫t-ɛt‍∫Rd‍Gα,δt-r,x-z2dzdr<Cɛ1-∑i=1d1/αi.By Lemma 17,(114)E(Jɛ,2(t,x))≤C∫t-ɛt‍ɛ∑i=1dhi-1/αi×E∫t-ɛt‍∫Rd‍Dr,zur1,z12dz1dr1dr≤Cɛ2-∑i=1d1/αi+∑i=1dhi-1/αi.Therefore, for each fixed ɛ0>0,(115)P∫0T‍∫RdDr,zut,x2dzdr>0≥sup⁡ɛ∈(0,ɛ0)P(C(Jɛ,1(t,x)-Jɛ,2(t,x))>0)≥sup⁡ɛ∈(0,ɛ0)P(Jɛ,2(t,x)<Cɛ1-∑i=1d1/αi)≥1-inf⁡ɛ∈0,ɛ01Cɛ1-∑i=1d1/αiEJɛ,2t,x≥1-inf⁡ɛ∈(0,ɛ0)1Cɛ1+∑i=1dhi-1/αi=1.This completes the proof of the theorem.


## 6. Conclusions and Discussions

We have obtained a unique solution of the stochastic fractional heat equation driven by fractional noise. The Lyapunov exponent of the solution has been estimated. In addition, we have discussed the Hölder continuity of the solution on both space and time parameters. Moreover, the absolute continuity of the solution has been obtained. Our future work will consider its application issues in computer networks, such as network traffic of fractional noise type passing through network servers.



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*Source: 421705-2015-04-23.xml*