Some stochastic time-fractional diffusion equations with variable coefficients and time dependent noise

TL;DR

This paper establishes the existence and uniqueness of solutions for a class of stochastic time-fractional diffusion equations with variable coefficients and complex noise structures, advancing the mathematical understanding of such models.

Contribution

It introduces new results on mild solutions for stochastic time-fractional PDEs with variable coefficients and general Gaussian noise, extending previous work to broader noise types and fractional orders.

Findings

Proved existence and uniqueness of mild solutions.

Handled a wide class of Gaussian noises including fractional, Riesz, and Bessel kernels.

Extended the theory to fractional orders in (1/2, 1) and (1, 2).

Abstract

We prove the existence and uniqueness of mild solution for the stochastic partial differential equation $$\left(\partial^\alpha - \textit{B} \right) u(t,x)= u(t,x) \cdot \dot{W}(t,x),$$ where $$\alpha \in (1/2, 1)\cup(1, 2);$$ $\textit{B}$ is an uniform elliptic operator with variable coefficients and $\dot W$ is a Gaussian noise general in time with space covariance given by fractional, Riesz and Bessel kernel.