Fractional diffusion in Gaussian noisy environment

TL;DR

This paper investigates the existence and uniqueness of solutions to fractional diffusion equations driven by Gaussian noise, focusing on the interplay between fractional derivatives, elliptic operators, and fractional Gaussian noise.

Contribution

It establishes conditions on the fractional order and Hurst parameters ensuring unique square integrable solutions to the stochastic PDE.

Findings

Derived existence and uniqueness conditions for solutions.

Identified parameter ranges for fractional order and Hurst parameters.

Provided mathematical framework for fractional diffusion in noisy environments.

Abstract

We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic partial equations of the following form: $D_t^\alpha u(t, x)=\textit{B}u+u\cdot W^H$, where $D_t^\alpha$ is the fractional derivative of order $\alpha$ with respect to the time variable $t$, $\textit{B}$ is a second order elliptic operator with respect to the space variable $x\in\mathbb{R}^d$, and $W^H$ a fractional Gaussian noise of Hurst parameter $H=(H_1, \cdots, H_d)$. We obtain conditions satisfied by $\alpha$ and $H$ so that the square integrable solution $u$ exists uniquely .