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

Abstract

We consider the stochastic heat equation of the form where is the fractional noise, is a (pure jump) Lévy space-time white noise, is Laplacian, and is the fractional Laplacian generator on , and are measurable functions. We introduce the existence and uniqueness of the solution by the fixed point principle under some suitable assumptions.

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 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 where is Laplacian, is the fractional Laplacian generator on , is the fractional noise, and is a (pure jump) Lévy space-time white noise. We first state two assumptions.

Assumption 1. For each , there exists a constant such that for all and .

Assumption 2. For some , we have The structure of this paper is as follows. In Section 2, we briefly present some basic notations and preliminaries on the pseudo-differential operator , 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 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 , 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

Consider a symmetric -stable motion with and an independent standard Brownian motion on . Then, the process is a diffusion such that its transition density function satisfies for all and , and moreover is the fundamental solution of equation The transition density function is also called the heat kernel of the operator . Denote for all and . For the heat kernel , we have the following estimates (see, for examples, Chen et al. [18], Kolokoltsov [21], and Bass and Levin [22]): for all and some constants , where for . In this paper, we only consider the case .

2.2. Lévy Space-Time White Noise

Let be a complete probability space with a usual filtration and let be two arbitrary measure spaces; is a -finite measure defined on for . Following, for example, Ikeda and Watanabe [23] or Truman and Wu [16], we denote which is called a Poisson random measure on , if, for all and , where .

In particular, when , we define the compensating -martingale measure for all with .

For any -predictable integrand which satisfies for some , we can define the stochastic integral which is a square integrable -martingale with the quadratic variation process

For the Poisson random measure and its compensating martingale measure , we can define the Radon-Nikodym derivatives for . A pure jump Lévy space-time white noise has the following structure: for some such that and , where are some measurable functions.

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

Proposition 3. Let be -predictable and satisfy (15). Denote by the integral process then for any and , there exists a constant such that

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

Assumption 4. For , the mappings satisfy, respectively,

2.3. Fractional Noises

Let denote a class of bounded Borel sets in and . Assume that is a centered Gaussian family of random variables with the covariance for , where denotes the Lebesgue measure of the set .

Let be the set of step functions on and let be the Hilbert space defined as the closure of with respect to the scalar product Then, the mapping is an isometry between and the linear space generated by , and moreover, the mapping can be extended to . This isometry is denoted by and is called the Wiener integral with respect to . Define the kernel by for , where is a normalization constant given as follows: Consider a linear operator defined by Then, the operator gives an isometry from to , and we find that (see, for example, Nualart [20] and Tindel et al. [7]) the processdefines a space-time white noise. Moreover, one can show that for . In particular, when the kernel can be rewritten as The following result follows from Mémin et al. [25].

Proposition 5. For one has

3. Existence and Uniqueness of the Solution

Let a filtered complete probability space be given as in the previous section. In this section, we will study the existence and uniqueness of the solution to the stochastic equation where is Laplacian, is the fractional Laplacian generator on , is the fractional noise, and 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 of . An -adapted process is a solution to (35) ifIn order to show the main theorem, we need the following lemma.

Lemma 6. Let and such that Define the operator byfor and . Then, for all , is a bounded linear operator from to . Specifically, we have the following:
(1) when , we have (2) when , we have (3) when and , we have

Proof. Clearly, we have for all and . It follows that for all and . Combining this with Minkowski’s inequality, (9), and Young’s inequality, we see that which gives case (1), and similarly, we can obtain case (2). Let us consider case (3).
For , we denote and denotes the complement of . We then see that for . It follows that This proves case (3) and the lemma follows.

Let be the space of all -valued -adapted processes . For fixed , define a functional on by for . Then, is a norm on and forms a Banach space. Consider the next integrals: for and and define the operator with .

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

Theorem 7. Let . Then, under Assumptions 1, 2, and 4, (35) admits a unique Walsh-mild solution such that for all , , and .

Based on the fixed point principle on the set , in order to prove the theorem, it is enough to prove the following two statements:(1)under Assumptions 1, 2, and 4, for and ;(2)under Assumptions 1, 2, and 4, the operator is a contraction on . In other words, there exists a constant such that  for .

Proof of Statement (1). Given , from (9), (46), Assumption 2, and Young’s inequality for , we have So .
Consider and take . It follows from Lemma 6 and Assumption 1 that for all , which gives .
For , by Proposition 5, we deduce that Moreover, similar to the proof of Lemma 6, we haveThis gives .
For , by Lemma 6 with and Assumption 4, it follows thatFinally, let us estimate . From Assumption 4, Lemma 6 with , and Proposition 3, it follows that Thus, we have showed that the operators , defined by (48) map to itself. On the other hand, in some same ways as in estimates (52)–(57), one can show that when sufficiently large. This completes the proof.

Proof of Statement (2). Suppose and are initials of -adapted random fields such that . We start with estimating . Note that for by (3) and Lemma 6 with . We get that where , which implies that with by choosing large enough.
Next we consider . We have that Thus, a similar procedure as above implies that is a contraction on by taking larger enough.
We finally consider . By the generalized B-D-G inequality (21) in Proposition 3, similar to (57), we can see that with by choosing large enough.
To sum up, we have shown that is a contraction on for large enough and statement (2) follows.

Remark 8. From the proof above, one can see that Theorem 7 is also true for .

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

Dengfeng Xia, Litan Yan, and Weiyin Fei carried out the mathematical studies, participated in the sequence alignment, drafted the manuscript, and participated in the design of the study and performed proof of results. All authors read and approved the final manuscript.

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).