Space-time fractional stochastic partial differential equations with L\'evy Noise

TL;DR

This paper studies non-linear space-time fractional stochastic heat equations driven by Lévy noise, proving existence, uniqueness, and growth behavior of solutions, extending classical results to fractional and jump noise settings.

Contribution

It introduces new existence and uniqueness results for space-time fractional SPDEs with Lévy noise, extending classical parabolic SPDE theory to fractional derivatives and jump processes.

Findings

Existence and uniqueness of mild solutions are established.

Solutions exhibit exponential growth under linear noise coefficient growth.

Nonexistence results are shown when the noise coefficient grows faster than linearly.

Abstract

We consider non-linear time-fractional stochastic heat type equation $$\frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg]$$ and $$\frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{N }(t,x,h)\bigg]$$ in $(d+1)$ dimensions, where $\alpha\in (0,2]$ and $d<\min\{2,\beta^{-1}\}\alpha$, $\nu>0$, $\partial^\beta_t$ is the Caputo fractional derivative, $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable process, $I^{1-\beta}_t$ is the fractional integral operator, ${N}(t,x)$ are Poisson random measure with $\tilde{N}(t,x)$ being the compensated Poisson random measure. $\sigma:{\mathbb{R}}\to{\mathbb{R}}$ is a Lipschitz continuous function. We prove existence and uniqueness of mild solutions to this equation. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in "M. Foondun and D. Khoshnevisan. Intermittence and nonlinear parabolic stochastic partial differential equations. \emph{ Electron. J. Probab.} {\bf14} (2009), 548--568" and " J. B. Walsh. An Introduction to Stochastic Partial Differential Equations, \'Ecoled'\'et\'e de Probabilit\'es de Saint-Flour, XIV|1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, (1986), 265--439". Under the linear growth of $\sigma$, we show that the solution of the time fractional stochastic partial differential equation follows an exponential growth with respect to the time. We also show the nonexistence of the random field solution of both stochastic partial differential equations when $\sigma$ grows faster than linear.

Equations

$$\frac{\partial^{\beta}u}{\partial t^{\beta}}+\nu(-\Delta)^{\alpha/2}u=I^{1-\beta}_{t}\bigg{[}\int_{\mathbb{R}^{d}}\sigma(u(t,x),h)\stackrel{{\scriptstyle\cdot}}{{\tilde{N}}}(t,x,h)\bigg{]}$$

$$\frac{\partial^{\beta}u}{\partial t^{\beta}}+\nu(-\Delta)^{\alpha/2}u=I^{1-\beta}_{t}\bigg{[}\int_{\mathbb{R}^{d}}\sigma(u(t,x),h)\stackrel{{\scriptstyle\cdot}}{{N}}(t,x,h)\bigg{]}$$

$$\frac{\partial^{\beta}u}{\partial t^{\beta}}+\nu(-\Delta)^{\alpha/2}u=I^{1-\beta}_{t}\bigg{[}\int_{\mathbb{R}^{d}}\sigma(u(t,x),h)\stackrel{{\scriptstyle\cdot}}{{\tilde{N}}}(t,x,h)\bigg{]}$$

$$\frac{\partial^{\beta}u}{\partial t^{\beta}}+\nu(-\Delta)^{\alpha/2}u=I^{1-\beta}_{t}\bigg{[}\int_{\mathbb{R}^{d}}\sigma(u(t,x),h)\stackrel{{\scriptstyle\cdot}}{{N}}(t,x,h)\bigg{]}$$

$${\partial }_t^{\beta }f(t)=\frac{1}{\Gamma (1-\beta )}\frac{d}{dt}{\int }_0^t(t-s{)}^{-\beta }(f(s)-f(0))ds,$$

$${\partial }_t^{\beta }{G}_t(x)=-\nu (-\Delta {)}^{\alpha /2}{G}_t(x).$$

$$\widehat{p_{X(s)}}(\xi )=e^{-s\nu |\xi {|}^{\alpha }},$$

$$f_{E_t}(x)=t{\beta }^{-1}{x}^{-1-1/\beta }{g}_{\beta }(t{x}^{-1/\beta }),$$

$$G_t(x)={\int }_0^{\infty }{p}_{{}_{X(s)}}(x)f_{E_t}(s)ds.$$

$${\int }_{{\mathbb{R}}^d}(1\wedge |h{|}^2)\mu (\mathrm{d}h)<\infty .$$

$$u(t,x)={\int }_{{\mathbb{R}}^d}G(t,x-y)u_0(y)\mathrm{d}y+{\int }_0^t{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}G(t-s,x-y)\sigma (u(s,y),h)\tilde{N}(\mathrm{d}s\mathrm{d}x\mathrm{d}h).$$

$$\sigma (0,h)=0\; and\; |\sigma (x,h)-\sigma (y,h)|\le J(h)Li{p}_{\sigma }|x-y|.$$

$$|\sigma (x,h)|\ge L\overline{J}(h)|x|.$$

$${\int }_{{\mathbb{R}}^d}\overline{J}(h{)}^2\mu (\mathrm{d}h)\ge \kappa ,$$

$$\underset{x\in {\mathbb{R}}^d}{\sup }\mathbb{E}|u(t,x){|}^2\le c_1{e}^{c_{2}t}for\; allt>0.$$

$$\underset{x\in {\mathbb{R}}^d}{\inf }\mathbb{E}|u(t,x){|}^2\ge c_1{e}^{c_{2}t}for\; allt>0.$$

$$\overline{\gamma }(2)\ge {{\rm Y}}^{-1}(\frac{1}{\kappa L^2})>0,$$

$${\int }_0^{\infty }{e}^{-{\rho }_{\ast }t}\mathbb{E}(|u_t(x){|}^2)\mathrm{d}t<\infty \text{ for all }x\in {\mathbb{R}}^d.$$

$$|\sigma (x,h)|\ge L\overline{J}(h)|x{|}^{\rho },$$

$$|\sigma (x,h)|\ge L\overline{J}(h)|x|.$$

$$\underset{x\in {\mathbb{R}}^d}{\inf }\mathbb{E}|u(t,x)|\ge c_1{e}^{c_{2}t}for\; allt>0.$$

$$|\sigma (x,h)|\ge L\overline{J}(h)|x{|}^{\lambda },$$

$${\int }_{{\mathbb{R}}^d}{G}_t^2(x)dx=C^{\ast }{t}^{-\beta d/\alpha }$$

$${\Phi }_t(x,h)=X{1}_{(a,b]}(t)\varphi (x,h)for\; allt\ge 0\; andx\in {\mathbb{R}}^d$$

$$\int h\Phi N(\mathrm{d}s\mathrm{d}x\mathrm{d}h)=X{\int }_{(a,b]\times {\mathbb{R}}^d\times {\mathbb{R}}^d}{h}_t(x,h)\varphi (x,h)N(\mathrm{d}s\mathrm{d}x\mathrm{d}h)$$

$${\int }_0^T{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}\mathbb{E}|X(s,x,h)|\mathrm{d}s\mathrm{d}x\mu (\mathrm{d}h)<\infty ,$$

$${\int }_0^T{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}X(s,x,h)N(\mathrm{d}s\mathrm{d}x\mathrm{d}h)$$

$$\begin{array}{rl}& \mathbb{E}{\int }_0^T{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}X(s,x,h)N(\mathrm{d}s\mathrm{d}x\mathrm{d}h)\\ & ={\int }_0^T{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}\mathbb{E}X(s,x,h)\mathrm{d}s\mathrm{d}x\mu (\mathrm{d}h).\end{array}$$

$${\int }_0^T{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}\mathbb{E}|X(s,x,h){|}^2\mathrm{d}s\mathrm{d}x\mu (\mathrm{d}h)<\infty ,$$

Full text

[TABLE]

and

[TABLE]

in $(d+1)$ dimensions, where $\alpha\in(0,2]$ and $d<\min\{2,\beta^{-1}\}\alpha$, $\nu>0$, $\partial^{\beta}_{t}$ is the Caputo fractional derivative, $-(-\Delta)^{\alpha/2}$ is the generator of an isotropic stable process, $I^{1-\beta}_{t}$ is the fractional integral operator, ${N}(t,x)$ are Poisson random measure with $\tilde{N}(t,x)$ being the compensated Poisson random measure. $\sigma:{\mathbb{R}}\to{\mathbb{R}}$ is a Lipschitz continuous function. We prove existence and uniqueness of mild solutions to this equation. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in [12, 26]. Under the linear growth of $\sigma$, we show that the solution of the time fractional stochastic partial differential equation follows an exponential growth with respect to the time. We also show the nonexistence of the random field solution of both stochastic partial differential equations when $\sigma$ grows faster than linear.

Keywords: Time-fractional stochastic partial differential equations; fractional Duhamel’s principle; Caputo derivatives; Walsh isometry

1. Introduction

Fractional calculus has received plenty of attention because its wide application in the field of physics, chemistry, finance and etc. In [14], we notice that many natural phenomena do not fit into the relatively simple description of diffusion developed by Einstein a century ago, such as the forage of food for animals in the forest, the transport of electrons in amorphous semiconductors in an electric field, the travel times of contaminants in groundwater and the proteins diffuse across cell membranes. Some of these phenomena follow models like Lévy flight, a fractal random walk, or composed of self-similar jumps. Mathematicians have been aware of fractional derivatives for over 300 years, but, like the Pareto distribution that has no mean value, these derivatives only find their ways into the physical sciences due to the relatively recent observations of anomalous diffusion: see, for example, [15, 22].

Stochastic partial differential equations (SPDE) have been studied in mathematics and various sciences as well; see, for example, Khoshnevisan [16] for a long list of references. The area of SPDEs is interesting to mathematicians because it contains a lot of hard open problems. However, not much have been done for equations driven by discontinuous noise even though this situation has started to change recently, see, for example, [5] and references therein.

In this paper we consider the following two time fractional stochastic partial differential equations (TSPDE),

[TABLE]

and

[TABLE]

where the initial condition $u_{0}$ is measurable and bounded, $-(-\Delta)^{\alpha/2}$ is the fractional Laplacian with $\alpha\in(0,2]$, $\stackrel{{\scriptstyle\cdot}}{{\tilde{N}}}(t,x)$ is a compensated Poisson noise with $x\in{\mathbb{R}}^{d}$, and $\dot{N}(t,x)$ is a compensated Poisson noise. Caputo fractional derivative $\partial^{\beta}_{t}$ of order $\beta\in(0,1)$, is defined by

[TABLE]

where $\Gamma(\beta):=\int^{\infty}_{0}t^{\lambda-1}e^{-t}dt$ is the Gamma function.

The meaning of the above fractional derivative of $f$ at time $t$ depends on the whole history of $f(s)$ on $(0,t)$ with the nearest past affecting the present more. (See [10]). The fractional diffusion equation $\partial^{\beta}_{t}u=\Delta u$ with $0<\beta<1$ has been widely used to model the anomalous diffusion exhibiting subdiffusive behavior due to the particle sticking and trapping phenomena (see e.g.[18]). We will prove the existence and uniqueness of the mild solution of equation (1.1) and (1.2) under the Lipschitz condition for $\sigma$. We will also discuss the existence of the finite energy solution and the blow-up and non-existence of the solution for both equations under some specific conditions. This paper is an extension of the results in the papers [24] and [23]. Here we consider the fractional time derivative and Poisson type noise. Also, Lévy noise ${\tilde{N}}(t,x,h)$ or $N(t,x,h)$ has better modeling characteristics than white noise in financial engineering[8], [9], signal detection [25], and other areas. It can capture some large moves and unpredictable events.

Let $G_{t}(x)$ be the fundamental solution of the fractional heat type equation

[TABLE]

$G_{t}(x)$ is the transition density function of $X(E_{t})$, where $X$ is an isotropic $\alpha$-stable Lévy process in ${\mathbb{R}}^{d}$ and $E_{t}$ is the first passage time of a $\beta$-stable subordinator $D=\{D_{r},\,r\geq 0\}$, or the inverse stable subordinator of index $\beta$: see, for example, Bertoin [6] for properties of these processes, Baeumer and Meerschaert [3] for more on time fractional diffusion equations, and Meerschaert and Scheffler [19] for properties of the inverse stable subordinator $E_{t}$.

Let $p_{{X(s)}}(x)$ and $f_{E_{t}}(s)$ be the density of $X(s)$ and $E_{t}$, respectively. Then the Fourier transform of $p_{{X(s)}}(x)$ is given by

[TABLE]

and

[TABLE]

where $g_{\beta}(\cdot)$ is the density function of $D_{1}.$ The function $g_{\beta}(u)$ (cf. Meerschaert and Straka [20]) is infinitely differentiable on the entire real line, with $g_{\beta}(u)=0$ for $u\leq 0$.

By conditioning, we have

[TABLE]

We define a Poisson random measure (or non-compensated Poisson random measure), $N:=\Sigma_{i\geq 1}\delta_{(T_{i},X_{i},Z_{i})}$ on ${\mathbb{R}}_{+}\times{{\mathbb{R}}^{d}}\times{{\mathbb{R}}^{d}}$ defined on a probability space$(\Omega,\mathcal{F},P)$ with intensity measure $\mathrm{d}t\mathrm{d}x\mu(\mathrm{d}h)$. Throughout this paper we assume that $\mu$ is a Lévy measure on ${\mathbb{R}}^{d}$, which satisfies the following

[TABLE]

Then we set $\tilde{N}(\mathrm{d}s\mathrm{d}x\mathrm{d}h)=N(\mathrm{d}s\mathrm{d}x\mathrm{d}h)-\mathrm{d}s\mathrm{d}x\mu(\mathrm{d}h)$ and call $\tilde{N}$ the compensated Poisson Random measure. In this paper we study the existence and uniqueness of the solution to (1.1) under global Lipchitz conditions on $\sigma$, using the white noise approach of [26].

We say that a random field $\{u(t,x)\}_{x\in{\mathbb{R}}^{d},t>0}$ is a mild solution of equation (1.1) if a.s., the following is satisfied

[TABLE]

For more explanation of mild solutions about Cauchy problem, please refer to [1]. We also refer to Mijena and Nane [23] for the use of time fractional Duhamel’s principle in obtaining the mild solutions.

For the existence and uniqueness of solutions to (1.1) we need the following condition on $\sigma$.

Condition 1.1**.**

There exists a non-negative function $J$ and a finite positive constant $Lip_{\sigma}$, such that for all $x,y,h\in{\mathbb{R}}^{d}$, we have

[TABLE]

The function $J$ is assumed to satisfy the following integrability condition, $\int_{{\mathbb{R}}^{d}}J(h)^{2}\mu(dh)\leq K$, where $K$ is some positive finite constant.

For the existence and uniqueness of solutions to (1.2) we need the following condition on $\sigma$.

Condition 1.2**.**

There exists a non-negative function $J$ and a finite positive constant $Lip_{\sigma}$, such that for all $x,y,h\in{{\mathbb{R}}^{d}}$, we have

[TABLE]

The function $J$ is assumed to satisfy the following integrability condition, $\int_{{\mathbb{R}}^{d}}J(h)\mu(dh)\leq K$, where $K$ is some positive finite constant.

The fractional integral of the noise term in equations (1.1) and (1.2) are not merely used to get a simple integral solution. A physical important reason to take the fractional integral of the noise in these equations: Apply the fractional derivative of order $1-\beta$ to both sides of these equations to see the forcing function, in the traditional units $x/t$: see, for example, Meerschaert et al [21]. In this paper the authors work on a deterministic time fractional equation with an external force, but the same physical principle should apply for the stochastic equations too.

We now briefly give an outline of this paper. We adapt the methods of proofs of the results in [23] with many crucial nontrivial changes. We state the main results of the paper in Section 2. We give some preliminary results in Section 3. Moment estimates for time increments and spatial increments of the solution are given in 4. The main result in this section is Proposition 4.1 and Proposition 4.2 under Lipschitz conditions of $\sigma$. In Sections 5 and 6, we prove the main results of the paper under some conditions of $\sigma$. We also give the behavior of the growth of the moments of the solutions when $\sigma$ is growing linearly. In addition, we also show that under faster than linear growth of $\sigma$, there is no finite energy solution for equation (1.1) with the compensated Poisson noise, and no random field solution for both equations.

2. Statement of main results

Our first existence and uniqueness result is the following theorem.

Theorem 2.1**.**

Let $d<\min\{2,\gamma^{-1}\}\alpha$. If $u_{0}$ is measurable and bounded, then there exists a unique random field solution to equation (1.1) under Condition 1.1.

Next, we show that a result of the growth of the second moment of the solution to equation (1.1) under condition of the linear growth of $\sigma$.

Condition 2.1**.**

There exists a positive function $\overline{J}$ and a constant L such that for all $x,h\in{\mathbb{R}}^{d}$ , we have

[TABLE]

The function $\overline{J}$ is assumed to satisfy the following integrability condition,

[TABLE]

where $\kappa$ is some positive constant.

The next result proves the intermittency property of the solution of equation (1.1).

Theorem 2.2**.**

Let $d<\min\{2,\gamma^{-1}\}\alpha$. Suppose that Condition 1.1 holds and $u_{0}$ is bounded above, which means there is a positive number $\eta_{1}$ such that $\eta_{1}:=\sup_{x\in{\mathbb{R}}^{d}}u_{0}(x)$, then the solution $u$ of equation (1.1) satisfies

[TABLE]

*where $c_{1}:=\eta_{1}^{2}$ and $c_{2}$ depends on $\alpha,\beta,d,\kappa$ and $Lip_{\sigma}$ in Condition 1.1.

Similarly, if Condition 2.1 holds and $u_{0}$ is bounded below, which means there is a positive number $\eta_{2}$ such that $\eta_{2}:=\inf_{x\in{\mathbb{R}}^{d}}u_{0}(x)$, then the solution $u$ of equation (1.1) satisfies*

[TABLE]

where $c_{1}:=\eta_{2}^{2}$ and $c_{2}$ depends on $\alpha,\beta,d,\kappa$ and $L$ in Condition 2.1.

If $\{X_{t}\}_{t\geq 0}$ is a Lévy process with characteristic function ${\mathbb{E}}\exp(i\xi X_{t})=\exp(-t\psi(\xi))$, for all $t>0$, $\xi\in{\mathbb{R}}^{d}$ where $\psi(\xi)$ is the characteristic exponent. When $\{X_{t}\}_{t\geq 0}$ is a symmetric $\alpha$-stable process, the characteristic exponent is $\psi(\xi)=\nu|\xi|^{\alpha}$, corresponding to the fractional Laplacian generator $-\nu(-\Delta)^{\alpha/2}$. Define $\Upsilon(\gamma):=\frac{1}{2\pi}\int_{{\mathbb{R}}^{d}}\frac{d\xi}{\gamma+2Re\psi(\xi)}\hbox{ for all }\gamma\in(0,\infty).$

Theorem 2.3**.**

Suppose that the assumption of Theorem 2.1 are in force, $\beta=1$, and that Condition 2.1 holds. If the initial function is bounded below, then

[TABLE]

where $\Upsilon^{-1}(t):=\sup\{\lambda>0:\Upsilon(\lambda)>t\}$.

This theorem is an extension of the corresponding result in [12] to SPDEs with Lévy noise.

We will establish the non-existence of finite energy solutions when $\sigma$ grows faster than linear.

A random field $u$ is a finite energy solution to the fractional stochastic heat equation (1.1) when $u\in\cup_{\gamma>0}\mathcal{L}^{\gamma,2}$ and there exists $\rho_{*}>0$ such that

[TABLE]

Condition 2.2**.**

There exist constants $L,\rho$ and a positive function $\overline{J}$ such that $\rho>1$ and for all $x,h\in{{\mathbb{R}}^{d}}$, we have

[TABLE]

where the function $\overline{J}$ is the same as in Condition 2.1.

Theorem 2.4**.**

Let let $d<\min\{2,\gamma^{-1}\}\alpha$. If the initial condition $u_{0}(x)$ is bounded below, then there is no finite energy solution to equation (1.1) under Condition 2.2.

Under the same condition, there is also no random field solution to equation (1.1).

Theorem 2.5**.**

Let $d<\min\{2,\gamma^{-1}\}\alpha$. If the initial condition $u_{0}(x)$ is bounded below, then there is no random field solution to equation(1.1) under Condition 2.2 .

Bao and Yuan [7] studied the finite time blow-up in $L^{p}$-norm of stochastic reaction-diffusion equations with jumps within a bounded domain. Li et al. [17] considered the blow-up in $L^{p}$-norm for a class of Lévy noise driven SPDEs.

In the remainder of the section we will state some properties of the solution to equation (1.2). First, we will present a similar theorem about the existence and uniqueness of the equation (1.2).

Theorem 2.6**.**

There exists a unique random field solution to the Equation (1.2) under Condition 1.2.

If the growth of $\sigma$ is linear, the next theorem shows that the solution of (1.2) grows exponentially.

Condition 2.3**.**

There exists a positive function $\overline{J}$ and a constant L such that for all $x,h\in{\mathbb{R}}^{d}$, we have

[TABLE]

The function $\overline{J}$ is assumed to satisfy the following integrability condition $\int_{{\mathbb{R}}^{d}}\overline{J}(h)dh\geq\kappa$, where $\kappa$ is some positive constant.

Theorem 2.7**.**

Suppose that Condition 1.2 and Condition 2.3 hold. If the initial condition $u_{0}(x)$ is bounded below, then the solution to equation (1.2) satisfies

[TABLE]

where $c_{1}:=\inf_{x\in R^{d}}|u(0,x)|$ and $c_{2}$ depends on $\alpha,\beta,d,\kappa$ and $L$ in Condition 2.3.

Condition 2.4**.**

There exists a constant $L,\lambda$ and a positive function $\overline{J}$, such that $\lambda>1$ and for all $x,h\in{{\mathbb{R}}^{d}},$ we have

[TABLE]

where the function $\overline{J}$ are the same as in Condition 2.3.

Theorem 2.8**.**

*If the initial condition $u_{0}(x)$ is bounded below, then there is no random field solution to equation(1.2) under Condition 2.4. *

3. Preliminaries

In this section, we give some preliminary results that will be needed in the remaining sections of the paper.

We first have the following lemma from [23].

Lemma 3.1** (Lemma 1 in Mijena and Nane [23] ).**

For $d<2\alpha,$

[TABLE]

where $C^{\ast}=\frac{(\nu)^{-d/\alpha}2\pi^{d/2}}{\alpha\Gamma(\frac{d}{2})}\frac{1}{(2\pi)^{d}}\int_{0}^{\infty}z^{d/\alpha-1}(E_{\beta}(-z))^{2}dz.$

Next we define the stochastic integrals with respect to Poisson random measures by giving the definition of simple random field.

Using the filtration of $\mathcal{F}_{t}=\sigma\{N(t),t\geq 0\}$, we say a random field $\Phi$ is* elementary* if it has the following representation:

[TABLE]

for some $0\leq a\leq b<\infty$, where $X$ is a $\mathcal{F}_{a}$ measurable random variable in $L^{2}(\Omega)$, and $\phi$ is non-random, bounded and measurable. It is natural to define the stochastic integral

[TABLE]

for any $h\in L^{2}({\mathbb{R}}_{+},{\mathbb{R}}^{d},{\mathbb{R}}^{d})$.

A random field $\Phi$ is simple if there exist elementary random fields $\Phi^{(1)},...,\Phi^{(n)}$ with disjoint support such that $\int h\Phi\mathrm{d}N=\Sigma_{i=1}^{n}\int h\Phi^{(i)}\mathrm{d}N$.

(See [16] for detailed definition based on the white noise.)

With this notion, we have the following sequence of definitions.

Definition 3.1**.**

Suppose that $X(s,x,h)$ is a predictable process such that

[TABLE]

then

[TABLE]

is well-defined and satisfies the following isometry

[TABLE]

We then have the following definition.

Definition 3.2**.**

Suppose that $X(s,x,h)$ is a predictable process such that

[TABLE]

then

[TABLE]

is well-defined and satisfies the following isometry

[TABLE]

We are now ready to state the precise meaning of the solutions. We define the mild solution of equation (1.1) first.

Definition 3.3**.**

We say that a random field $\{u(t,x)\}_{t>0,x\in{\mathbb{R}}^{d}}$ is a mild solution of equation (1.1) if a.s., the following is satisfied

[TABLE]

Since we are mainly interested in the second moment of the solution of equation (1.1), and we say that if $\{u(t,x)\}_{t>0,x\in{\mathbb{R}}^{d}}$ satisfies the following condition

[TABLE]

for all $T>0$, then $\{u(t,x)\}_{t>0,x\in{\mathbb{R}}^{d}}$ is a $random$ $field$ $solution$ to the equation (1.1).

For any $\gamma>0$, define the following norm

[TABLE]

Definition 3.4**.**

We denote by $\mathcal{L}^{\gamma,2}$ the completion of the space of all simple random field in the norm $\||u||_{2,\gamma}.$

Switching to the solution of equation (1.2), we similarly define the mild solution of equation (1.2).

Definition 3.5**.**

We say that a random field $\{u(t,x)\}_{t>0,x\in{\mathbb{R}}^{d}}$ is a mild solution of equation (1.2) if a.s.,the following is satisfied

[TABLE]

We are interested in the first moment of the solution of equation(1.2), and we say that if $\{u(t,x)\}_{t>0,x\in{\mathbb{R}}^{d}}$ satisfies the following condition

[TABLE]

for all $T>0$, then $\{u(t,x)\}_{t>0,x\in{\mathbb{R}}^{d}}$ is a $random$ $field$ $solution$ to the equation (1.2).

Define the following norm for any $\gamma>0$

[TABLE]

Definition 3.6**.**

We denote by $\mathcal{L}^{\gamma,1}$ the completion of the space of all simple random fields in the norm $\||u||_{1,\gamma}.$

We also quote the following propositions that will be needed in the proof of our main results.

Proposition 3.1** (Proposition 2.12 in Foondun and Nane [13]).**

Let $\rho>0$ and suppose $f(t)$ is nonnegative, locally integrable functions satisfying

[TABLE]

where $c_{1}$ is some positive number. Then, we have

[TABLE]

for some positive constants $c_{2}$ and $c_{3}$.

Proposition 3.2** (Proposition 2.13 in Foondun and Nane [13]).**

Let $\rho>0$ and suppose $f(t)$ is nonnegative, locally integrable functions satisfying

[TABLE]

where $c_{1}$ is some positive number. Then, we have

[TABLE]

for some positive constants $c_{2}$ and $c_{3}$.

Proposition 3.3** (Proposition 2.12 in Asogwa et al. [2]).**

Let $\theta>0$. Suppose $h$ is a non-negative function satisfying the following non-linear integral inequality,

[TABLE]

where $C$, $D$ and $\gamma$ are positive numbers. Then for any $C>0$ there exists $t_{0}>0$ such that $h(t)=\infty$ for all $t\geq t_{0}$.

4. Estimates on moments of the increments of the solution

In this section we are going to prove a stochastic Young’s inequality for both compensated Poisson integrals and non-compensated Poisson integrals. These inequalities are very crucial for proving our main theorems on existence and uniqueness of solutions.

Now we set $(P_{t}u_{0})(x):=\int_{{\mathbb{R}}^{d}}G(t,x-y)u_{0}(y)\mathrm{d}y$ and

[TABLE]

Next, we prove a stochastic Young’s Inequality for equation (1.1) with compensated Poisson noise.

Proposition 4.1**.**

Suppose that $d<min\{2,\beta^{-1}\}\alpha$, Condition 1.1 holds, $u$ admits a predictable random field solution of equation (1.1) and $||u||_{2,\gamma}<\infty$ for $\gamma>0$. Then there exists some positive constant $C^{**}$ such that $||\mathcal{A}u||_{2,\gamma}\leq C^{**}||u||_{2,\gamma}$, where $C^{**}=Lip_{\sigma}\sqrt{\frac{{C^{*}K}\Gamma(1-\frac{\beta d}{\alpha})}{\gamma^{1+\frac{\beta d}{\alpha}}}}$.

Proof.

By applying the Walsh isometry for compensated Poisson integrals in equation (3.3), Lemma 3.1, and Condition 1.1 we get

[TABLE]

Let $r=\gamma s$, then

[TABLE]

Therefore $||\mathcal{A}u||_{2,\gamma}\leq C^{**}||u||_{2,\gamma}$, where $C^{**}=Lip_{\sigma}\sqrt{\frac{{C^{*}K}\Gamma(1-\frac{\beta d}{\alpha})}{\gamma^{1+\frac{\beta d}{\alpha}}}}$. ∎

We then present a corollary of the above stochastic Young’s inequality by some substitutions.

Corollary 4.1**.**

Suppose that $d<\min\{2,\beta^{-1}\}\alpha$,$\gamma>0$, and Condition 1.1 holds. For any predictable random field solutions $u$ and $v$ of (1.1) satisfying $||u||_{2,\gamma}+||v||_{2,\gamma}<\infty,$ we have $||\mathcal{A}u-\mathcal{A}v||_{2,\gamma}\leq C^{**}||u-v||_{2,\gamma}$, where $C^{**}=Lip_{\sigma}\sqrt{\frac{{C^{*}K}\Gamma(1-\frac{\beta d}{\alpha})}{\gamma^{1+\frac{\beta d}{\alpha}}}}$.

Proof.

Applying the proof of Proposition 4.1 to $u-v$, we have

[TABLE]

Take the square root, it yields $||\mathcal{A}u-\mathcal{A}v||_{2,\gamma}\leq C^{**}||u-v||_{2,\gamma}$, where $C^{**}=Lip_{\sigma}\sqrt{\frac{{C^{*}K}\Gamma(1-\frac{\beta d}{\alpha})}{\gamma^{1+\frac{\beta d}{\alpha}}}}$.

∎

There is also a corresponding stochastic Young’s inequality for equation (1.2) with non-compensated Poisson noise. We define the following operator first

[TABLE]

Proposition 4.2**.**

Suppose that u is predictable random field solution of equation (1.2) such that $||u||_{1,\gamma}<\infty$, then under Condition 1.2, we have

[TABLE]

Proof.

Using the isometry (3.2) we have

[TABLE]

Mutiply both side by $e^{-\gamma t}$, it yields

[TABLE]

∎

We also obtain a corollary of the above stochastic Young’s inequality for Equation (1.2) with non-compensated Poisson noise.

Corollary 4.2**.**

Suppose that Condition 1.2 holds. For any predictable random field solutions $u$ and $v$ of (1.2) satisfying $||u||_{1,\gamma}+||v||_{1,\gamma}<\infty$ for all $\gamma>0$, we have

[TABLE]

Proof.

The proof is similar to the proof of Corollary 4.1. ∎

5. Proof of results for the compensated Poisson noise

Proof of Theorem 2.1.

We use Picard iteration to show existence of solutions. Let $u^{(0)}(t,x):=u_{0}(x)$, and for all $n\geq 0$

[TABLE]

and

[TABLE]

Then we have

[TABLE]

Since by Proposition 4.1, $||\mathcal{A}u||_{2,\gamma}\leq C^{**}||u||_{2,\gamma},\text{for }\gamma>0$ it yields

[TABLE]

We can find a constant $L_{2}$ depending only on $Lip_{\sigma},\alpha,\beta,d$ such that $\gamma:=(L_{2}K)^{\frac{\alpha}{\alpha+\beta d}}$ satisfying

[TABLE]

and let $\theta=\sup_{x\in{\mathbb{R}}^{d}}|u_{0}(x)|$

then

[TABLE]

Hence, $||u^{(n+1)}||_{2,\gamma}\in\cup_{\gamma>0}\mathcal{L}^{\gamma,2},$ i.e.

[TABLE]

Therefore,we obtain

[TABLE]

Next, we use the Banach fixed point theorem to show the existence of the solution. By the stochastic Young’s inequality, we have

[TABLE]

For any $m\geq n>0$,

[TABLE]

so $||u^{(m)}-u^{(n)}||_{2,\gamma}\to 0$ as $n\to\infty$. Since $\mathcal{L}^{\gamma,2}$ is complete, then there exists $u\in\mathcal{L}^{\gamma,2}$, such that $u^{(n)}\to u$ in the norm sense.

The uniqueness of the the solution to Equation (1.1) follows easily the above argument by picking $u$ and $v$ as two solutions to the equation, and by using Corollary 4.1. ∎

Proof of Theorem 2.2.

We begin with the isometry (3.3),

[TABLE]

By assumption, Condition 1.1, and Lemma 3.1 we have

[TABLE]

By letting $F(t)=\sup_{x\in{{\mathbb{R}}^{d}}}{\mathbb{E}}|u(t,x)|^{2}$, the above inequality becomes:

[TABLE]

Since $\beta d/\alpha<1$, then $1-(\beta d/\alpha)>0$. By setting $c_{1}=\eta_{1}^{2},\kappa^{{}^{\prime}}=KC^{*}L^{2},\rho=1-(\beta d/\alpha)$ in Proposition 3.1, we obtain the required result.

Now let us move on to the proof of the second part of the theorem.

Similarly, by Condition 2.1, and Lemma 3.1 we have

[TABLE]

By letting $F(t)=\inf_{x\in{{\mathbb{R}}^{d}}}{\mathbb{E}}|u(t,x)|^{2}$, it yields

[TABLE]

By Proposition 3.2, we obtain the required result for the lower bound for all $t>0$.

∎

Proof of Theorem 2.3.

Following Theorem 2.1, there exists a unique mild solution to Equation 1.1 when $\beta=1$ , that is

[TABLE]

Take the second moment together with Condition 2.1, we have

[TABLE]

Set $\eta:=\inf_{x\in{\mathbb{R}}^{d}}u_{0}(x)$ and $\tilde{t}=t-s$, we obtain

[TABLE]

Let $F_{\lambda}(x):=\int_{0}^{\infty}e^{-\lambda t}E|u(t,x)|^{2}\mathrm{d}t,$ then we obtain:

[TABLE]

Where

[TABLE]

and

[TABLE]

where $p(t,x)$ is the density of the $\alpha$-stable process whose generator is the $-(-\Delta)^{\alpha/2}$, since $\beta=1.$

Then by using the same line of ideas as in the proof of Theorem 2.7 in [12] we can show that

[TABLE]

Form this we see that $F_{\lambda}(x)=\infty$ as long as $\kappa L^{2}\Upsilon(\lambda)\geq 1.$ ∎

Proof of Theorem 2.4.

[TABLE]

Let $\eta:=\inf_{x\in{{\mathbb{R}}^{d}}}u_{0}(x)$, by Condition 2.2 and Jensen’s inequality, we have

[TABLE]

Let $I(t):=\inf_{y\in{{\mathbb{R}}^{d}}}{\mathbb{E}}|u(s,y)|^{2}$, by Lemma 3.1, we have

[TABLE]

Define the Laplace transform of $I$ as

[TABLE]

It is easy to see that

[TABLE]

for all $\theta>0$, where $C_{1}=C^{*}\Gamma(1-\beta d/\alpha)$. It follows that

[TABLE]

Multiply both sides by $\theta$ and use Jensen’s inequality to get

[TABLE]

for all $\theta>0$. It follows that $\theta\tilde{I}(\theta)\geq\eta^{2}>0$, and hence $\theta\tilde{I}(\theta)>0$ for all $\theta>0.$ Moreover,

[TABLE]

Hence,

[TABLE]

For $0<\theta<\theta_{0}:=(\kappa L^{2}C_{1}\eta^{2\rho-2})^{1-\beta d/\alpha}$, we will show that $\tilde{I}(\theta)=\infty.$

Under the assumption that $0<\theta\leq\theta_{0}$, the constant

[TABLE]

With the recursive argument, we have

[TABLE]

for any positive integer $n$. So $\theta\tilde{I}(\theta)=\infty,$ which means $\tilde{I}(\theta)=\infty$ for a specified range of $\theta$. Since ${I}(\theta_{0})=\infty$, then $I(t)\geq ce^{\theta_{0}t}$ for large $t$. So for $t$ large enough, we have

[TABLE]

Since $\theta_{0}\rho t/2>0$, using the inequality $1-e^{-x}>\frac{x}{x+1}$ for $x>-1$, we have

[TABLE]

And there exists a $t^{*}$, such that $\frac{\theta_{0}\rho t/2}{1+\theta_{0}\rho t/2}>\frac{\theta_{0}\rho t/2}{2}.$ Hence,

[TABLE]

for $t$ large enough. Let $\theta_{1}=\theta_{0}\rho$, then we have $\tilde{I}(\theta)=\infty.$ Otherwise, there is no $\theta$ less than $\theta_{1}$ such that $\tilde{I}(\theta)<\infty.$

On the other hand, if we assume there is a finite energy solution, we have $\tilde{I}(\theta_{*})<\infty$ for some $\theta_{*}>0$. Hence

[TABLE]

for $\theta\geq\theta_{\ast}$. That means $\tilde{I}(\theta)<\infty$ for all $\theta\geq\theta_{\ast}.$ But this contradicts the above argument. Therefore there is no finite energy solution. ∎

Proof of Theorem 2.5.

We begin with the isometry (3.2),

[TABLE]

Let $\eta:=\inf_{x\in{{\mathbb{R}}^{d}}}u_{0}(x)$. By Condition 2.2 and Jensen’s inequality, we have

[TABLE]

Let $F(t)=\inf_{x\in{{\mathbb{R}}^{d}}}{\mathbb{E}}|u(t,x)|^{2}$, then the above is reduced to be

[TABLE]

Using Lemma 3.1 yields

[TABLE]

Since $\eta^{2}$,$\kappa C^{*}L^{2}>0$, and $\rho>1,\beta d/\alpha>0$, then from Proposition 3.3, we know that there exists a $t_{0}$ such that ${\mathbb{E}}|u(t,x)|^{2}=\infty$ for all $t\geq t_{0}$, which means

[TABLE]

So $u(t,x)$ will blow up in finite time and there is no random field solution to equation (1.1). ∎

6. Proof of results for the Poisson noise

Proof of Theorem 2.6.

We use Picard iteration to show existence of solutions. Let $u^{(0)}(t,x):=u_{0}(x)$, and for all $n\geq 0$

[TABLE]

and

[TABLE]

Then we have

[TABLE]

By Propostion 4.2, $||\mathcal{B}u||_{1,\gamma}\leq\frac{KLip_{\sigma}}{\gamma}||u||_{1,\gamma}\ \hbox{for }\ \gamma>0$. This yields

[TABLE]

We can find a constant $L_{3}$– depending only on $Lip_{\sigma},K$–such that $\gamma:=L_{3}KLip_{\sigma}$ satisfying

[TABLE]

and let $\theta=\sup_{x\in{\mathbb{R}}^{d}}|u_{0}(x)|$.

Then

[TABLE]

so we have $||u^{(n+1)}||_{1,\gamma}\in\cup_{\gamma>0}\mathcal{L}^{\gamma,1}.$ i.e.

[TABLE]

Hence $\sup_{x\in{\mathbb{R}}^{d}}{\mathbb{E}}|u^{(n+1)}(t,x)|\leq L_{4}e^{\gamma t},t>0.$

Next, we use the Banach fixed point theorem to show the existence of the solution. By the Corollary 4.2, we have

[TABLE]

Following the last part of the proof of Theorem 2.1, we obtain the existence and uniqueness of the solution to equation (1.2). ∎

Proof of Theorem 2.7.

We use Walsh isometry (3.2)

[TABLE]

Let $\eta:=\inf_{x\in{\mathbb{R}}^{d}}u_{0}(x)$. By Condition 2.3, we have

[TABLE]

Let $f(t):=\inf_{x\in{\mathbb{R}}^{d}}{\mathbb{E}}|u(t,x)|$, the above yields

[TABLE]

So we get the intended result by Proposition 3.2. ∎

Proof of Theorem 2.8.

[TABLE]

Let $\eta:=\inf_{x\in{\mathbb{R}}^{d}}u_{0}(x)$ and $\eta>0$, we have $(P_{t}u_{0})(x)>\eta$. By Condition 2.4 and Jensen’s inequality, we have

[TABLE]

Let $f(t):=\inf_{x\in{\mathbb{R}}^{d}}{\mathbb{E}}|u(t,x)|$, then we have

[TABLE]

Since $\eta$, $\kappa C^{*}L,\beta d/\alpha>0$, and $\lambda>1$, then there is no random field solution to Equation (1.2).

∎