Harnack Inequality and Strong Feller Property for Stochastic Fast-Diffusion Equations

TL;DR

This paper establishes Harnack inequalities and the strong Feller property for stochastic fast-diffusion equations, overcoming technical challenges due to weaker dissipativity by employing Sobolev-Nash inequalities.

Contribution

It extends the analysis of Harnack inequalities and strong Feller properties to stochastic fast-diffusion equations using Sobolev-Nash inequalities, which was previously unexplored.

Findings

Harnack inequalities are proved for stochastic fast-diffusion equations.

The strong Feller property is established under weaker dissipativity conditions.

Concrete examples illustrate the applicability of the main results.

Abstract

This paper presents analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation possesses weaker dissipativity than the porous medium one does, some technical difficulties appear in the study. As a compensation to the weaker dissipativity condition, a Sobolev-Nash inequality is assumed for the underlying self-adjoint operator in applications. Some concrete examples are constructed to illustrate the main results.