Space-time fractional diffusions in Gaussian noisy environment

TL;DR

This paper investigates fractional space-time stochastic PDEs driven by Gaussian noise, establishing existence, uniqueness, and moment bounds of solutions using Fox H-functions, and revealing new properties of fundamental solutions.

Contribution

It introduces new analytical results for fractional stochastic PDEs with Gaussian noise, including solution properties and fundamental solution characteristics.

Findings

Existence and uniqueness of solutions established.

Moment bounds for solutions derived.

New properties of fundamental solutions identified.

Abstract

This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables $\left(\partial^\beta + \frac{\nu}{2} (-\Delta)^{\alpha/2} \right) u(t,x)= \lambda u(t,x) \dot{W}(t,x)$, where $\dot W$ is a general Gaussian noise and $\beta\in (1/2, 2)$, $\alpha\in (0, 2]$. The existence and uniqueness of the solution, the moment bounds of the solution are obtained by using the fundamental solutions of the corresponding deterministic counterpart represented by the Fox H-functions. Along the way, we obtain some new properties of the fundamental solutions.