Mild Solutions for Time--Fractional Stochastic Nonlocal Diffusion Equations

TL;DR

This paper investigates mild solutions for a time-space nonlocal stochastic diffusion equation driven by white noise, highlighting the impact of fractional derivatives, spatial dimension, and diffusion coefficients on solution existence.

Contribution

It provides explicit solution formulas, characterizes existence conditions for mild solutions, and extends known results for local fractional stochastic heat equations to nonlocal cases.

Findings

Explicit solution formula involving Mittag-Leffler functions

Existence of mild solutions depends on fractional order, dimension, and coefficients

Numerical simulations show subdiffusive behavior and memory effects

Abstract

We study a time--space nonlocal diffusion equation driven by additive time--space white noise, where the time derivative is the Caputo derivative of order $\alpha\in(0,2)$. The model couples local diffusion with a nonlocal convolution operator generated by a radial probability density, thus incorporating memory effects and long-range spatial interactions. For Dirac initial data, we derive an explicit solution formula in the space of tempered distributions, decomposing the solution into a deterministic part and a stochastic convolution kernel expressed through Mittag--Leffler functions. Our main contribution is a sharp characterization of the existence of mild solutions in terms of $\alpha$, the spatial dimension $N$, and the coefficients of the local and nonlocal diffusion terms. In particular, when the Laplacian term is absent, no mild solution exists, whereas for $\lambda>0$ the admissible regimes depend critically on $(\alpha,N)$, extending and sharpening the known results for purely local fractional stochastic heat equations. Numerical simulations illustrate the evolution of the mean and variance and emphasize subdiffusive spreading and memory effects.