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

Abstract

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

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Dengfeng Xia 1; 2 and Litan Yan 1; 3 and Weiyin Fei 2

Academic Editor:Maria L. Gandarias

1, College of Information Science and Technology, Donghua University, 2999 North Renmin Rd., Songjiang, Shanghai 201620, China

2, School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, Anhui 241000, China

3, Department of Mathematics, College of Science, Donghua University, 2999 North Renmin Rd., Songjiang, Shanghai 201620, China

Received 23 December 2016; Revised 20 March 2017; Accepted 21 March 2017; 13 April 2017

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

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 operators _Δα +_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 [figure omitted; refer to PDF] 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 each T>0, there exists a constant C>0 such that [figure omitted; refer to PDF] for all (t,x,y)∈[0,T]×R×R and u,u¯∈R.

Assumption 2.

For some p≥2, we have [figure omitted; refer to PDF] 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 operator Δ+_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 Operator Δ+_Δα

Consider a symmetric α-stable motion ^Xα ={^Xtα , t≥0} with α∈(0,2) and an independent standard Brownian motion B on ^Rd . Then, the process [figure omitted; refer to PDF] is a diffusion such that its transition density function ^G(α) (x,t) satisfies [figure omitted; refer to PDF] for all t≥0 and z∈^Rd , and moreover ^G(α) (x,t) is the fundamental solution of equation [figure omitted; refer to PDF] The transition density function ^G(α) is also called the heat kernel of the operator Δ+_Δα . Denote [figure omitted; refer to PDF] for 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]): [figure omitted; refer to PDF] for 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 [figure omitted; refer to PDF] which is called a Poisson random measure on (_U1 ,B(_U1 ),_ν1 ), if, for all _A1 ∈B(_U1 ), _A2 ∈B(_U2 ) and n∈N∪{∞}, [figure omitted; refer to PDF] where N={0,1,2,...}.

In particular, when _U1 =[0,∞)×R, we define the compensating {_Ft }-martingale measure [figure omitted; refer to PDF] for all (t,_A1 ,_A2 )∈[0,∞)×B(R)×B(_U2 ) with _ν1 ((0,t)×_A1 )_ν2 (_A2 )<∞.

For any (_Ft )-predictable integrand [straight phi]:[0,∞)×R×_U2 ×Ω[arrow right]R which satisfies [figure omitted; refer to PDF] for some (_A1 ,_A2 )∈B(R)×B(_U2 ), we can define the stochastic integral [figure omitted; refer to PDF] which is a square integrable (_Ft)t≥0 -martingale with the quadratic variation process [figure omitted; refer to PDF]

For the Poisson random measure N and its compensating martingale measure Q, we can define the Radon-Nikodym derivatives [figure omitted; refer to PDF] for (t,x,y)∈[0,∞)×R×_U2 . A pure jump Lévy space-time white noise has the following structure: [figure omitted; refer to PDF] for some _A0 ∈_U2 such that _ν2 (_U2 \_A0 )<∞ and _∫A0^z2_ν2 (dz)<+∞, where _g1 ,_g2 :[0,∞)×R×_U2 [arrow right]R are some measurable functions.

Next we quote the following B-D-G inequality (see, for example, [24] or [10]).

Proposition 3.

Let [straight phi]:[0,∞)×R×_U2 ×Ω[arrow right]R be (_Ft)t≥0 -predictable and satisfy (15). Denote by Y the integral process [figure omitted; refer to PDF] then for any T>0 and p>1, there exists a constant _Cp,T >0 such that [figure omitted; refer to PDF]

In order to handle (1) we claim also the following assumptions.

Assumption 4.

For p≥2, the mappings [figure omitted; refer to PDF] satisfy, respectively, [figure omitted; refer to PDF]

2.3. Fractional Noises

Let _Bb (R) denote a class of bounded Borel sets in R and H∈(1/2,1). Assume that [figure omitted; refer to PDF] is a centered Gaussian family of random variables with the covariance [figure omitted; refer to PDF] for s,t∈[0,T], _A1 ,_A2 ∈_Bb (R), where |_A1 | denotes the Lebesgue measure of the set _A1 ∈_Bb (R).

Let S 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] and is called the Wiener integral with respect to ^WH . Define the kernel _KH (t,s) by [figure omitted; refer to PDF] for H∈(0,1), where _cH >0 is a normalization constant given as follows: [figure omitted; refer to PDF] Consider a linear operator ^{KH[low *]} :S[...]^L2 ([0,T]×R) defined by [figure omitted; refer to PDF] Then, the operator ^{KH[low *]} 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 [figure omitted; refer to PDF] defines a space-time white noise. Moreover, one can show that [figure omitted; refer to PDF] for t∈[0,T], A∈_Bb (R). In particular, when H>1/2 the kernel _KH can be rewritten as [figure omitted; refer to PDF] The following result follows from Mémin et al. [25].

Proposition 5.

For H>1/2 one has [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] 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[arrow right]R is a solution to (35) if [figure omitted; refer to PDF] In order to show the main theorem, we need the following lemma.

Lemma 6.

Let 1<=r<+∞ and 1<=^{p[variant prime]} <=p<+∞ such that [figure omitted; refer to PDF] Define the operator O by [figure omitted; refer to PDF] for v∈^L1 ([0,T],^{Lp[variant prime]} (R)) and G∈(^G(α) ,(^G(α))2 ,(∂/∂y)^G(α) ). Then, for all t∈[0,T], O is a bounded linear operator from ^L1 ([0,T],^{Lp[variant prime]} (R)) to ^L∞ ([0,T],^Lp (R)). Specifically, we have the following:

(1) when G=^G(α) , we have [figure omitted; refer to PDF]

(2) when G=(^G(α))2 , we have [figure omitted; refer to PDF]

(3) when G=(∂/∂y)^G(α) and r=1, we have [figure omitted; refer to PDF]

Proof.

Clearly, we have [figure omitted; refer to PDF] for all t>s>0 and 1<α<2. It follows that [figure omitted; refer to PDF] for all t>s>0 and 1<α<2. Combining this with Minkowski's inequality, (9), and Young's inequality, we see that [figure omitted; refer to PDF] which gives case (1), and similarly, we can obtain case (2). Let us consider case (3).

For t>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 [figure omitted; refer to PDF] for t>s>0. It follows that [figure omitted; refer to PDF] This proves case (3) and the lemma follows.

Let B be the space of all ^Lp (R)-valued _Ft -adapted processes u(t,·_)0<=t<=T :[0,T]×R×Ω[arrow right]R. For fixed η>0, define a functional _(·)B on B by [figure omitted; refer to PDF] for u∈B. Then, _(·)B is a norm on B and (B,_(·)B ) forms a Banach space. Consider the next integrals: [figure omitted; refer to PDF] for u∈B and (t,x)∈[0,T]×R and define the operator [figure omitted; refer to PDF] with u∈B.

In this section, our main object is to expound and to prove the next theorem.

Theorem 7.

Let 1/2<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 [figure omitted; refer to PDF] 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 Assumptions 1, 2, and 4, ^Bui ∈B for i=0,1,2,3,4 and u∈B;

(2) under Assumptions 1, 2, and 4, the operator B(u) is a contraction on B. In other words, there exists a constant c∈(0,1) such that [figure omitted; refer to PDF]

: for u,v∈B.

Proof of Statement (1).

Given η>0, from (9), (46), Assumption 2, and Young's inequality for 1/p=1+1/p-1, we have [figure omitted; refer to PDF] So ^Bu0 ∈B.

Consider ^Bu1 and take r=1. It follows from Lemma 6 and Assumption 1 that [figure omitted; refer to PDF] for all u∈B, which gives ^Bu1 (t,x)∈B.

For ^Bu2 , by Proposition 5, we deduce that [figure omitted; refer to PDF] Moreover, similar to the proof of Lemma 6, we have [figure omitted; refer to PDF] This gives ^Bu2 (t,x)∈B.

For ^Bu3 , by Lemma 6 with 1/r=1-2/p+1/p=1-1/p and Assumption 4, it follows that [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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).

Suppose _u0 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 [figure omitted; refer to PDF] for ^{p[variant prime]} =p/3 by (3) and Lemma 6 with 1/r=1/p-1/^{p[variant prime]} +1=1-2/p. We get that [figure omitted; refer to PDF] where κ=p(1-r)/2r(1-p), which implies that [figure omitted; refer to PDF] with c∈(0,1) by choosing η>0 large enough.

Next we consider ^Bu3 . We have that [figure omitted; refer to PDF] Thus, a similar procedure as above implies that ^Bu3 is a contraction on B by taking η>0 larger enough.

We finally consider ^Bu4 . By the generalized B-D-G inequality (21) in Proposition 3, similar to (57), we can see that [figure omitted; refer to PDF] with c∈(0,1) by choosing η>0 large enough.

To sum up, we have shown that B(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 Theorem 7 is also true for H=1/2.

Acknowledgments

The work is supported and sponsored by National Natural Science Foundation of China (Grant nos. 11571071, 71271003, and 71571001), Natural Science Foundation of Anhui Province (Grant no. 1608085MA02), and the Foundation for Young Talents in College of Anhui Province (Grant no. gxyq2017014).

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