Space-time fractional diffusion of Riesz-Bessel type on regular bounded open domains

TL;DR

This paper investigates fractional diffusion equations with stochastic noise on bounded domains, establishing conditions for solutions and analyzing their regularity properties using Mittag-Leffler functions and eigenvalue asymptotics.

Contribution

It provides new conditions for defining weak Gaussian solutions to space-time fractional diffusion equations driven by white noise on bounded domains.

Findings

Conditions for weak-sense Gaussian solutions are derived.

Mean-square H"older continuity of solutions is established.

Asymptotic behavior of solutions is characterized using Mittag-Leffler functions.

Abstract

Fractional (in time and in space) evolution equations defined on Dirichlet regular bounded open domains, driven by fractional integrated in time Gaussian spatiotemporal white noise, are considered here. Sufficient conditions for the definition of a weak-sense Gaussian solution, in the mean-square sense, are derived. The temporal, spatial and spatiotemporal H\"older continuity, in the mean-square sense, of the formulated solution is obtained, under suitable conditions, from the asymptotic properties of the Mittag-Leffler function, and the asymptotic order of the eigenvalues of a fractional polynomial of the Dirichlet negative Laplacian operator on such bounded open domains.