# Mixed Fractional Heat Equation Driven by Fractional Brownian Sheet and Lévy Process

Authors: Dengfeng Xia; Litan Yan; Weiyin Fei
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

We consider the stochastic heat equation of the form∂u/∂t=(Δ+Δα)u+(∂f/∂x)(t,x,u)+σ(t,x,u)L˙+W˙H, where W˙H is the fractional noise, L˙ is a (pure jump) Lévy space-time white noise, Δ is Laplacian, and Δα=-(-Δ)α/2 is the fractional Laplacian generator on R, and f,σ:[0,T]×R×R→R are measurable functions. We introduce the existence and uniqueness of the solution by the fixed point principle under some suitable assumptions.


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## Body

## 1. Introduction

Stochastic calculus of fractional Brownian motion (in short, fBm) naturally led to the study of stochastic partial differential equations (in short, SPDEs) driven by it, and the study of such SPDEs constitutes an important research direction in probability theory and stochastic analysis, and many interesting researches have been done. The motivation comes from wide applications of fBm. We refer, among others, to Duncan et al. [1], Hu [2], Jiang et al. [3, 4], Liu and Yan [5], Sobczyk [6], Tindel et al. [7], Mishura et al. [8], and the references therein. On the other hand, as is well known, SPDEs driven by Lévy noise constitute a very important research direction and many significant researches have been carried out. We mention the works of Bo et al. [9, 10], Shi and Wang [11], Mueller [12], Chen et al. [13], Løkka et al. [14], and Truman and Wu [15, 16]. However, it is not sufficient to study the mixed heat equation with fractional and Lévy noises.It is important to note that the increasing interest to study the pseudo-differential operatorsΔα+Δβ is motivated by its applications to fluid dynamic traffic model, statistical mechanics, and heat conduction in materials with memory and also because they can be employed to approach nonlinear conservation laws. Therefore, it seems interesting to handle the mixed fractional heat equation. In the recent paper of Xia and Yan [17], they introduced only the existence and uniqueness of the solution of a mixed fractional heat equation driven by a fractional Brownian sheet. As an extension, in the present paper, we consider the stochastic heat equation of the form (1)∂ut,x∂t=Δ+Δαut,x+∂f∂xt,x,ut,x+σt,x,ut,xL˙+W˙H,t≥0,x∈R,u0,x=u0x,x∈R,where Δ is Laplacian, Δα=-(-Δ)α/2 is the fractional Laplacian generator on R, W˙H is the fractional noise, and L˙ is a (pure jump) Lévy space-time white noise. We first state two assumptions.Assumption 1.
For eachT>0, there exists a constant C>0 such that (2)ft,x,y≤C1+y,(3)ft,x,u-ft,x,u¯+σt,x,u-σt,x,u¯≤Cu-u¯for all (t,x,y)∈[0,T]×R×R and u,u¯∈R.Assumption 2.
For somep≥2, we have (4)Eu0·pp<∞.The structure of this paper is as follows. In Section 2, we briefly present some basic notations and preliminaries on the pseudo-differential operator Δ+Δα, Lévy space-time white noise, and fractional noise. In Section 3, we study the existence and uniqueness of the Walsh-mild solution to (1).


## 2. Preliminaries

In this section, we briefly recall some basic results for Green function of the pseudo-differential operatorΔ+Δα and stochastic calculus associated with fractional Brownian sheet and Lévy space-time white noise. We refer to Chen et al. [18, 19], Shi and Wang [11], Nualart [20], and the references therein for more details. In this paper, the letter C, with or without subscripts, stands for a positive constant whose value is unimportant and which may change from location to location, even within a line; we also stress that it depends on some constants.


### 2.1. Pseudo-Differential OperatorΔ+Δα

Consider a symmetricα-stable motion Xα={Xtα,t≥0} with α∈(0,2) and an independent standard Brownian motion B on Rd. Then, the process (5)Yt≔Bt+Xtα,t≥0,is a diffusion such that its transition density function G(α)(x,t) satisfies (6)∫RdGαx,teizxdx=e-tzα+z2for all t≥0 and z∈Rd, and moreover G(α)(x,t) is the fundamental solution of equation (7)∂u∂t=Δ+Δαu.The transition density function G(α) is also called the heat kernel of the operator Δ+Δα. Denote (8)Gαs,y;t,x=Gαt-s,x-yfor all x,y∈Rd and s,t≥0. For the heat kernel G(α), we have the following estimates (see, for examples, Chen et al. [18], Kolokoltsov [21], and Bass and Levin [22]): (9)Gαs,y;t,x=Gαt-s,x-y≤C1t-s-d/2e-x-y2/C2t-s+t-s-d/2∧t-sx-yd+α,(10)∂Gα∂ts,y;t,x≤Ct-sGαs,y;t,x,(11)∂Gα∂ys,y;t,x≤Ct-s-1/α∧1x-yGαs,y;t,xfor all t>s>0,x,y∈Rd and some constants C,C1,C2>1, where a1∧a2≔min{a1,a2} for a1,a2∈R. In this paper, we only consider the case d=1.


### 2.2. Lévy Space-Time White Noise

Let(Ω,F,P) be a complete probability space with a usual filtration {Ft}t≥0 and let (Ui,B(Ui))(i=1,2) be two arbitrary measure spaces; νi is a σ-finite measure defined on (Ui,B(Ui)) for i=1,2. Following, for example, Ikeda and Watanabe [23] or Truman and Wu [16], we denote (12)N:U1,BU1,ν1×U2,BU2,ν2×Ω,F,P⟶N∪∞,which is called a Poisson random measure on (U1,B(U1),ν1), if, for all A1∈B(U1),A2∈B(U2) and n∈N∪{∞}, (13)Pω∈Ω:NA1,A2,ω=n=1n!ν1nA1ν2nA2e-ν1A1ν2A2,where N={0,1,2,…}.In particular, whenU1=[0,∞)×R, we define the compensating {Ft}-martingale measure (14)Qt,A1,A2≔N0,t×A1,A2-tν1A1ν2A2,for all (t,A1,A2)∈[0,∞)×B(R)×B(U2) with ν10,t×A1ν2A2<∞.For any(Ft)-predictable integrand φ:[0,∞)×R×U2×Ω→R which satisfies (15)E∫0t∫A1×A2φs,x,y2dsdxν2dy<∞,∀t∈0,∞for some (A1,A2)∈B(R)×B(U2), we can define the stochastic integral (16)∫0t+∫A1×A2φs,x,yQdy,dx,ds≔∫0t+∫A1×A2φs,x,yNdy,dx,ds-∫0t∫A1×A2φs,x,yν2dydxds,which is a square integrable (Ft)t≥0-martingale with the quadratic variation process (17)∫0·+∫A1×A2φs,x,yQdy,dx,dst=∫0t∫A1×A2φs,x,y2ν2dydxds.For the Poisson random measureN and its compensating martingale measure Q, we can define the Radon-Nikodym derivatives (18)N~dy,x,t≔Ndtdx,dydtdx,Q~dy,x,t≔Qdtdx,dydtdxfor (t,x,y)∈[0,∞)×R×U2. A pure jump Lévy space-time white noise has the following structure: (19)L˙x,t=∫A0g1t,x,yQ~dy,x,t+∫U2∖A0g2t,x,yN~dy,x,tfor some A0∈U2 such that ν2(U2∖A0)<∞ and ∫A0z2ν2(dz)<+∞, where g1,g2:[0,∞)×R×U2→R are some measurable functions.Next we quote the following B-D-G inequality (see, for example, [24] or [10]).Proposition 3.
Letφ:[0,∞)×R×U2×Ω→R be (Ft)t≥0-predictable and satisfy (15). Denote by Y the integral process (20)Yt≔∫0t+∫R×U2φs,y,zQdz,dy,ds,t≥0;then for any T>0 and p>1, there exists a constant Cp,T>0 such that (21)supt∈0,TEYtp≤Cp,T∫0T∫R×U2Eφs,y,zp2/pν2dzdydsp/2.In order to handle (1) we claim also the following assumptions.Assumption 4.
Forp≥2, the mappings (22)ψt,x=∫U2∖U0g2t,x,yν2dy,gt,x,y=g1t,x,y1A0y+g2t,x,y1U2∖A0ysatisfy, respectively, (23)supt∈0,Tψt,·pp<∞,supt∈0,T∫U2gt,·,y2ν2dyp/2p/2<∞.


### 2.3. Fractional Noises

LetBb(R) denote a class of bounded Borel sets in R and H∈(1/2,1). Assume that (24)WH=WH0,t×A1,t∈0,T,A1∈BbRis a centered Gaussian family of random variables with the covariance (25)EWH0,s×A1WH0,t×A2=A1∩A2RHs,t,RHs,t=12t2H+s2H-t-s2Hfor s,t∈[0,T],A1,A2∈Bb(R), where |A1| denotes the Lebesgue measure of the set A1∈Bb(R).LetS be the set of step functions on [0,T]×R and let H be the Hilbert space defined as the closure of S with respect to the scalar product (26)10,t×A1,10,s×A2H=A1∩A2RHt,s.Then, the mapping 1[0,t]×A1↦WH([0,t]×A1) is an isometry between S and the linear space generated by WH, and moreover, the mapping can be extended to H. This isometry is denoted by (27)ϕ⟼WHϕ=∫0T∫Rϕs,yWHds,dy,ϕ∈H,and is called the Wiener integral with respect to WH. Define the kernel KH(t,s) by (28)KHt,s=cHtsH-1/2t-sH-1/2+12-Hs1/2-H∫strH-3/2r-sH-1/2drfor H∈(0,1), where cH>0 is a normalization constant given as follows: (29)cH=2HΓ3/2-HΓH+1/2Γ2-2H.Consider a linear operator KH∗:S↦L2([0,T]×R) defined by (30)KH∗ψs,x=KHT,sψs,x+∫sTψt,x-ψs,x∂KH∂tt,sdt.Then, the operator KH∗ gives an isometry from H to L2([0,T]×R), and we find that (see, for example, Nualart [20] and Tindel et al. [7]) the process(31)WA1×0,t≔WHKH∗-110,t×A1defines a space-time white noise. Moreover, one can show that (32)WHA×0,t=∫0t∫AKHt,sWdy,dsfor t∈[0,T],A∈Bb(R). In particular, when H>1/2 the kernel KH can be rewritten as (33)KHt,s=H-12cHs1/2-H∫stuH-1/2u-sH-3/2du,KH∗ψ·,xt=∫tT∂KH∂ts,tψs,xds.The following result follows from Mémin et al. [25].Proposition 5.
ForH>1/2 one has (34)L1/H0,T×R⊂H.


## 2.1. Pseudo-Differential OperatorΔ+Δα

Consider a symmetricα-stable motion Xα={Xtα,t≥0} with α∈(0,2) and an independent standard Brownian motion B on Rd. Then, the process (5)Yt≔Bt+Xtα,t≥0,is a diffusion such that its transition density function G(α)(x,t) satisfies (6)∫RdGαx,teizxdx=e-tzα+z2for all t≥0 and z∈Rd, and moreover G(α)(x,t) is the fundamental solution of equation (7)∂u∂t=Δ+Δαu.The transition density function G(α) is also called the heat kernel of the operator Δ+Δα. Denote (8)Gαs,y;t,x=Gαt-s,x-yfor all x,y∈Rd and s,t≥0. For the heat kernel G(α), we have the following estimates (see, for examples, Chen et al. [18], Kolokoltsov [21], and Bass and Levin [22]): (9)Gαs,y;t,x=Gαt-s,x-y≤C1t-s-d/2e-x-y2/C2t-s+t-s-d/2∧t-sx-yd+α,(10)∂Gα∂ts,y;t,x≤Ct-sGαs,y;t,x,(11)∂Gα∂ys,y;t,x≤Ct-s-1/α∧1x-yGαs,y;t,xfor all t>s>0,x,y∈Rd and some constants C,C1,C2>1, where a1∧a2≔min{a1,a2} for a1,a2∈R. In this paper, we only consider the case d=1.


## 2.2. Lévy Space-Time White Noise

Let(Ω,F,P) be a complete probability space with a usual filtration {Ft}t≥0 and let (Ui,B(Ui))(i=1,2) be two arbitrary measure spaces; νi is a σ-finite measure defined on (Ui,B(Ui)) for i=1,2. Following, for example, Ikeda and Watanabe [23] or Truman and Wu [16], we denote (12)N:U1,BU1,ν1×U2,BU2,ν2×Ω,F,P⟶N∪∞,which is called a Poisson random measure on (U1,B(U1),ν1), if, for all A1∈B(U1),A2∈B(U2) and n∈N∪{∞}, (13)Pω∈Ω:NA1,A2,ω=n=1n!ν1nA1ν2nA2e-ν1A1ν2A2,where N={0,1,2,…}.In particular, whenU1=[0,∞)×R, we define the compensating {Ft}-martingale measure (14)Qt,A1,A2≔N0,t×A1,A2-tν1A1ν2A2,for all (t,A1,A2)∈[0,∞)×B(R)×B(U2) with ν10,t×A1ν2A2<∞.For any(Ft)-predictable integrand φ:[0,∞)×R×U2×Ω→R which satisfies (15)E∫0t∫A1×A2φs,x,y2dsdxν2dy<∞,∀t∈0,∞for some (A1,A2)∈B(R)×B(U2), we can define the stochastic integral (16)∫0t+∫A1×A2φs,x,yQdy,dx,ds≔∫0t+∫A1×A2φs,x,yNdy,dx,ds-∫0t∫A1×A2φs,x,yν2dydxds,which is a square integrable (Ft)t≥0-martingale with the quadratic variation process (17)∫0·+∫A1×A2φs,x,yQdy,dx,dst=∫0t∫A1×A2φs,x,y2ν2dydxds.For the Poisson random measureN and its compensating martingale measure Q, we can define the Radon-Nikodym derivatives (18)N~dy,x,t≔Ndtdx,dydtdx,Q~dy,x,t≔Qdtdx,dydtdxfor (t,x,y)∈[0,∞)×R×U2. A pure jump Lévy space-time white noise has the following structure: (19)L˙x,t=∫A0g1t,x,yQ~dy,x,t+∫U2∖A0g2t,x,yN~dy,x,tfor some A0∈U2 such that ν2(U2∖A0)<∞ and ∫A0z2ν2(dz)<+∞, where g1,g2:[0,∞)×R×U2→R are some measurable functions.Next we quote the following B-D-G inequality (see, for example, [24] or [10]).Proposition 3.
Letφ:[0,∞)×R×U2×Ω→R be (Ft)t≥0-predictable and satisfy (15). Denote by Y the integral process (20)Yt≔∫0t+∫R×U2φs,y,zQdz,dy,ds,t≥0;then for any T>0 and p>1, there exists a constant Cp,T>0 such that (21)supt∈0,TEYtp≤Cp,T∫0T∫R×U2Eφs,y,zp2/pν2dzdydsp/2.In order to handle (1) we claim also the following assumptions.Assumption 4.
Forp≥2, the mappings (22)ψt,x=∫U2∖U0g2t,x,yν2dy,gt,x,y=g1t,x,y1A0y+g2t,x,y1U2∖A0ysatisfy, respectively, (23)supt∈0,Tψt,·pp<∞,supt∈0,T∫U2gt,·,y2ν2dyp/2p/2<∞.


## 2.3. Fractional Noises

LetBb(R) denote a class of bounded Borel sets in R and H∈(1/2,1). Assume that (24)WH=WH0,t×A1,t∈0,T,A1∈BbRis a centered Gaussian family of random variables with the covariance (25)EWH0,s×A1WH0,t×A2=A1∩A2RHs,t,RHs,t=12t2H+s2H-t-s2Hfor s,t∈[0,T],A1,A2∈Bb(R), where |A1| denotes the Lebesgue measure of the set A1∈Bb(R).LetS be the set of step functions on [0,T]×R and let H be the Hilbert space defined as the closure of S with respect to the scalar product (26)10,t×A1,10,s×A2H=A1∩A2RHt,s.Then, the mapping 1[0,t]×A1↦WH([0,t]×A1) is an isometry between S and the linear space generated by WH, and moreover, the mapping can be extended to H. This isometry is denoted by (27)ϕ⟼WHϕ=∫0T∫Rϕs,yWHds,dy,ϕ∈H,and is called the Wiener integral with respect to WH. Define the kernel KH(t,s) by (28)KHt,s=cHtsH-1/2t-sH-1/2+12-Hs1/2-H∫strH-3/2r-sH-1/2drfor H∈(0,1), where cH>0 is a normalization constant given as follows: (29)cH=2HΓ3/2-HΓH+1/2Γ2-2H.Consider a linear operator KH∗:S↦L2([0,T]×R) defined by (30)KH∗ψs,x=KHT,sψs,x+∫sTψt,x-ψs,x∂KH∂tt,sdt.Then, the operator KH∗ gives an isometry from H to L2([0,T]×R), and we find that (see, for example, Nualart [20] and Tindel et al. [7]) the process(31)WA1×0,t≔WHKH∗-110,t×A1defines a space-time white noise. Moreover, one can show that (32)WHA×0,t=∫0t∫AKHt,sWdy,dsfor t∈[0,T],A∈Bb(R). In particular, when H>1/2 the kernel KH can be rewritten as (33)KHt,s=H-12cHs1/2-H∫stuH-1/2u-sH-3/2du,KH∗ψ·,xt=∫tT∂KH∂ts,tψs,xds.The following result follows from Mémin et al. [25].Proposition 5.
ForH>1/2 one has (34)L1/H0,T×R⊂H.


## 3. Existence and Uniqueness of the Solution

Let a filtered complete probability space(Ω,F,(Ft)t≥0,P) be given as in the previous section. In this section, we will study the existence and uniqueness of the solution to the stochastic equation (35)∂ut,x∂t=Δ+Δαut,x+∂f∂xt,x,ut,x+σt,x,ut,xL˙+W˙H,t≥0,x∈R,u0,x=u0x,x∈R,where Δ is Laplacian, Δα=-(-Δ)α/2 is the fractional Laplacian generator on R, W˙H is the fractional noise, and L˙ is a (pure jump) Lévy space-time white noise. Moreover, we assume also that Assumptions 1, 2, and 4 in Sections 1 and 2 hold.From Walsh [26], one can introduce a notation of Walsh-mild solution to (35) by using the heat kernel G(α)(s,y;t,x)=G(α)(t-s;x-y) of Δ+Δα. An Lp(Ω)Ft-adapted process u:[0,T]×R×Ω→R is a solution to (35) if(36)ut,x=∫RGαt,x-yu0ydy+∫0t∫RGαt-s,x-yWHds,dy+∫0t∫R∂Gα∂yt-s,x-yfs,y,us,ydyds+∫0t∫RGαt-s,x-yσs,y,us,yψs,ydyds+∫0t+∫R×U2Gαt-s,x-yσs,y,us,ygs,y,zQdy,dz,ds.In order to show the main theorem, we need the following lemma.Lemma 6.
Let1≤r<+∞ and 1≤p′≤p<+∞ such that (37)1r=1+1p-1p′.Define the operator O by(38)Ovt,x=∫0t∫RGt-s,x-yvs,ydyds,for v∈L1([0,T],Lp′(R)) and G∈Gα,(Gα)2,(∂/∂y)G(α). Then, for all t∈[0,T], O is a bounded linear operator from L1([0,T],Lp′(R)) to L∞([0,T],Lp(R)). Specifically, we have the following:
(1) whenG=G(α), we have (39)Ovt,·p=∫0t∫RGαt-s,·-yvs,ydydsp≤C∫0tt-s1-r/2rvs,·p′ds;
(2) whenG=(G(α))2, we have (40)Ovt,·p=∫0t∫RGαt-s,·-y2vs,ydydsp≤C∫0tt-s1-2r/2rvs,·p′ds;
(3) whenG=(∂/∂y)G(α) and r=1, we have (41)Ovt,·p=∫0t∫R∂Gα∂yt-s,·-yvs,ydydsp≤C∫0tt-s-1/2vs,·pds.Proof.
Clearly, we have(42)∫Rt-s-1/2e-y2/C2t-srdy=Cr-1/2t-s1-r/2,∫Rt-s-1/2∧t-sy1+αrdy≤Ct-s1-r/2for all t>s>0 and 1<α<2. It follows that (43)Gαt-s,·r=∫RGαt-s,yrdy1/r≤C∫Rt-s-1/2e-y2/C2t-s+t-s-1/2∧t-sy1+αrdy1/r≤Ct-s1-r/2rfor all t>s>0 and 1<α<2. Combining this with Minkowski’s inequality, (9), and Young’s inequality, we see that (44)Ovt,·p=∫0t∫RGαt-s,·-yvs,ydydsp≤C∫0t∫RGαt-s,·-yvs,ydypds≤C∫0tGαt-s,·∗vs,··pds≤C∫0tGαt-s,·rvs,·p′ds≤C∫0tt-s1-r/2rvs,·p′ds,which gives case (1), and similarly, we can obtain case (2). Let us consider case (3).
Fort>s>0, we denote D1=y∈R∣|y|<(t-s)3/2(1+α) and D¯1 denotes the complement of D1. We then see that (45)∂Gα∂·t-s,·1=∫R∂Gα∂yt-s,ydy≤C∫D1t-s1/αt-s-1/2e-y2/C2t-s+t-s-1/2dy+C∫D¯11yt-s-1/2e-y2/C2t-s+t-sy1+αdy≤Ct-s1/α-1/2∫D1e-y2/C2t-s+1dy+Ct-s-1/2∫D¯11ye-y2/C2t-s+t-s3/2y2+αdy≤Ct-s-1/2for t>s>0. It follows that (46)Ovt,·p=∫0t∫R∂Gα∂yt-s,·-yvs,ydydsp≤C∫0t∂Gα∂·t-s,·∗vs,··pds≤C∫0t∂Gα∂·t-s,·1vs,·pds≤C∫0tt-s-1/2vs,·pds.This proves case (3) and the lemma follows.LetB be the space of all Lp(R)-valued Ft-adapted processes u(t,·)0≤t≤T:[0,T]×R×Ω→R. For fixed η>0, define a functional ·B on B by (47)uB≔supt∈0,Te-ηtEut,·pp1/pfor u∈B. Then, ·B is a norm on B and (B,·B) forms a Banach space. Consider the next integrals: (48)Bu0t,x≔∫RGαt,x-yu0ydy,Bu1t,x≔∫0t∫R∂Gα∂yt-s,x-yfs,y,us,ydyds,Bu2t,x≔∫0t∫RGαt-s,x-yWHdy,ds,Bu3t,x≔∫0t∫RGαt-s,x-yσs,y,us,yψs,ydyds,Bu4t,x≔∫0t∫R×U2Gαt-s,x-yσs,y,us,ygs,y,zQdz,dy,ds,for u∈B and (t,x)∈[0,T]×R and define the operator (49)But,x≔∑i=04Buit,x,t,x∈0,T×Rwith u∈B.In this section, our main object is to expound and to prove the next theorem.Theorem 7.
Let1/2<H<1. Then, under Assumptions 1, 2, and 4, (35) admits a unique Walsh-mild solution u={u(t,x),(t,x)∈[0,T]×R} such that (50)supt∈0,TEut,·pp<∞for all T>0, α∈(1,2), and p≥4.Based on the fixed point principle on the set{u∈B:u(0)=u0}, in order to prove the theorem, it is enough to prove the following two statements:(1)
under Assumptions1, 2, and 4, Bui∈B for i=0,1,2,3,4 and u∈B;(2)
under Assumptions1, 2, and 4, the operator B(u) is a contraction on B. In other words, there exists a constant c∈(0,1) such that (51)Bu-BvB≤cu-vBforu,v∈B.Proof of Statement (1).
Givenη>0, from (9), (46), Assumption 2, and Young’s inequality for 1/p=1+1/p-1, we have (52)Bu0t,xp=∫RGαt,x-yu0ydyp≤CGαt,·∗u0··p≤CGαt,·1·u0·p≤Cu0·p<∞.So Bu0∈B.
ConsiderBu1 and take r=1. It follows from Lemma 6 and Assumption 1 that (53)EBu1t,xpp=E∫0t∫R∂Gα∂yt-s,x-yfs,y,us,ydydspp≤CE∫0t∂Gα∂·t-s,·1·fs,·,us,·pdsp≤CE∫0t1+us,·pdsp≤C1+sups∈0,TEus,·pp≤C1+us,·BP<∞for all u∈B, which gives Bu1(t,x)∈B.
ForBu2, by Proposition 5, we deduce that (54)EBu2t,xpp=E∫0t∫RGαt-s,x-yWHdy,dspp=∫RE∫0t∫RGαt-s,x-yWHdy,dspdx=∫RE∫0t∫RKH∗Gαt-·,x-·s,yWdy,dspdx≤C∫RKH∗Gαt-·,x-·,KH∗Gαt-·,x-·L20,T×Rp/2dx=C∫RGαt-·,x-·,Gαt-·,x-·Hp/2dx≤C∫RGαt-·,x-·L1/H0,T×Rpdx.Moreover, similar to the proof of Lemma 6, we have(55)Gαt-·,x-·L1/H0,T×Rp=∫0t∫RGαt-s,x-y1/HdydspH≤C∫0tt-sH-1/2HdspH<∞.This gives Bu2(t,x)∈B.
ForBu3, by Lemma 6 with 1/r=1-2/p+1/p=1-1/p and Assumption 4, it follows that(56)EBu3t,xpp=E∫0t∫RGαt-s,x-yσs,y,us,yψs,ydydspp≤CE∫0tt-s1-r/2r1+us,·p·ψt,·pdsp≤C∫0tt-s1-r/2rdsp1+u·Bpsupt∈0,Tψt,·pp≤Csupt∈0,Tψt,·pp·1+u·Bp.Finally, let us estimate Bu4(t,x). From Assumption 4, Lemma 6 with 1/r=2/p-4/p+1=1-2/p∈(0,1], and Proposition 3, it follows that (57)EBu4t,xpp≤C∫R∫0t∫R×U2ν2dzdyds·EGαt-s,x-yσs,y,us,ygs,y,zp2/pp/2≤C∫R∫0t∫R×U2Gαt-s,x-ygs,y,z2·E1+us,yp2/pν2dzdyds2/pdx=C∫0t∫RGαt-s,·-y2·∫U2gs,·,z2ν2dz1+Eus,yp2/pdydsp/2p/2≤C∫0tGαt-s,·2r∫U2gs,·,z2ν2dz1+Eus,·p2/pp/4dsp/2≤C∫0tt-s1-2r/2rdsp/2·supt∈0,T1+Eus,·p2/pp/2p/2·supt∈0,T∫U2gs,·,z2ν2dzp/2p/2≤C1+u·Bpsupt∈0,T∫U2gs,·,z2ν2dzp/2p/2<∞.Thus, we have showed that the operators Bui, i=0,1,2,3,4 defined by (48) map B to itself. On the other hand, in some same ways as in estimates (52)–(57), one can show that Bui∈B when η>0 sufficiently large. This completes the proof.Proof of Statement (2).
Supposeu0 and v0 are initials of (Ft)t≥0-adapted random fields u,v∈B such that u0=v0. We start with estimating Bu1. Note that (58)EBu1t,·-Bv1t,·pp=E∫0t∫R∂Gα∂yt-s,·-yfs,y,us,y-fs,y,vs,ydydspp≤CE∫0tt-s1-r/2rfs,y,us,y-fs,y,vs,ypdsp≤CE∫0tt-s1-r/2rEus,·-vs,·pdspfor p′=p/3 by (3) and Lemma 6 with 1/r=1/p-1/p′+1=1-2/p. We get that (59)Bu1t,·-Bv1t,·Bp=supt∈0,Te-ηtEBu1t,·-Bv1t,·pp≤Csupt∈0,TE∫0tt-s1-r/2re-ηt/pus,·-vs,·pdsp≤Csupt∈0,TE∫0te-ηsus,·-vs,·ppds∫0tt-s1-r/2re-ηt-s/pp/p-1dsp-1≤Csupt∈0,T∫0te-ηsEus,·-vs,·ppds∫0tt-s1-r/2re-ηt-s/pp/p-1dsp-1≤Cu-vBp∫0∞t-s1-r/2re-ηt-s/pp/p-1dsp-1=Cu-vBpp-1ηκ+1p-1∫0∞e-xxκdxp-1≤CTu-vBpp-1ηκ+1p-1Γκ+1p-1,where κ=p1-r/2r1-p, which implies that (60)Bu1t,·-Bv1t,·B≤Cp-1ηκ+1p-1/pΓκ+1p-1/p·u-vB≤cu-vBwith c∈(0,1) by choosing η>0 large enough.
Next we considerBu3. We have that (61)EBu3t,·-Bv3t,·pp=E∫0t∫RGαt-s,·-yψs,yσs,y,us,y-σs,y,vs,ydydspp≤CE∫0tt-s1-r/2rψs,·pσs,y,us,y-σs,y,vs,ypdsp≤Csupt∈0,Tψt,·ppE∫0tt-s1-r/2rσs,y,us,y-σs,y,vs,ypdsp.Thus, a similar procedure as above implies that Bu3 is a contraction on B by taking η>0 larger enough.
We finally considerBu4. By the generalized B-D-G inequality (21) in Proposition 3, similar to (57), we can see that (62)EBu4t,·-Bv4t,·Bp=supt∈0,Te-ηtEBu4t,·-Bv4t,·pp=supt∈0,Te-ηt∫RE∫0t+∫R×U2Gαt-s,x-ygs,y,z·σs,y,us,y-σs,y,vs,yQdz,dy,dspdx≤cu-vBp,with c∈(0,1) by choosing η>0 large enough.
To sum up, we have shown thatB(u) is a contraction on B for η>0 large enough and statement (2) follows.Remark 8.
From the proof above, one can see that Theorem7 is also true for H=1/2.



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*Source: 8059796-2017-04-13.xml*