Space-time fractional stochastic partial differential equations driven by L\'evy white noise

TL;DR

This paper studies space-time fractional stochastic PDEs driven by Lévy white noise, establishing existence and uniqueness of solutions in various function spaces under different noise conditions.

Contribution

It introduces new existence and uniqueness results for solutions of fractional stochastic PDEs driven by Lévy noise, including both Gaussian and pure jump cases.

Findings

Existence and uniqueness of $L^2$-valued local solutions with Gaussian noise.

Existence and uniqueness of $L^p$-valued local solutions with pure jump Lévy noise.

Conditions for global solutions under stronger assumptions.

Abstract

This paper is concerned with the following space-time fractional stochastic nonlinear partial differential equation \begin{equation*} \left(\partial_t^{\beta}+\frac{\nu}{2}\left(-\Delta\right)^{\alpha / 2}\right) u=I_{t}^{\gamma}\Big[ f(t,x,u)-\sum_{i=1}^{d} \frac{\partial}{\partial x_i} q_i(t,x,u)+ \sigma(t,x,u) F_{t,x}\Big] \end{equation*} for a random field $u(t,x):[0,\infty)\times\mathbb{R}^d \mapsto\mathbb{R}$, where $\alpha>0, \beta\in(0,2), \gamma\ge0, \nu>0, F_{t,x}$ is a L\'evy space-time white noise, $I_{t}^\gamma$ stands for the Riemann-Liouville integral in time, and $f,q_i,\sigma:[0,\infty)\times\mathbb{R}^d\times\mathbb{R} \mapsto\mathbb{R}$ are measurable functions. Under suitable polynomial growth conditions, we establish the existence and uniqueness of $L^2(\mathbb{R}^d)$-valued local solutions when the L\'evy white noise $F_{t,x}$ contains Gaussian noise component. Furthermore, for $p\in[1,2]$, we derive the existence and uniqueness of $L^p(\mathbb{R}^d)$-valued local solutions for the equation driven by pure jump L\'evy white noise. Finally, we obtain certain stronger conditions for the existence and uniqueness of global solutions.