Regularity of stochastic nonlocal diffusion equations

TL;DR

This paper establishes the regularity and Hölder continuity of solutions to stochastic nonlocal diffusion equations, providing explicit estimates and criteria, including for cases involving time-space white noise.

Contribution

It introduces new regularity results and explicit Hölder continuity criteria for solutions of stochastic nonlocal PDEs, applicable to time-space white noise.

Findings

Solutions are Hölder continuous with explicit indices.

Derived estimates for solutions in Sobolev and Hölder spaces.

Established criteria for Hölder continuity on bounded domains.

Abstract

In this paper, we are concerned with regularity of nonlocal stochastic partial differential equations of parabolic type. By using Companato estimates and Sobolev embedding theorem, we first show the H\"{o}lder continuity (locally in the whole state space $\mathbb{R}^d$) for mild solutions of stochastic nonlocal diffusion equations in the sense that the solutions $u$ belong to the space $C^{\gamma}(D_T;L^p(\Omega))$ with the optimal H\"{o}lder continuity index $\gamma$ (which is given explicitly), where $D_T:=[0,T]\times D$ for $T>0$, and $D\subset\mathbb{R}^d$ being a bounded domain. Then, by utilising tail estimates, we are able to obtain the estimates of mild solutions in $L^p(\Omega;C^{\gamma^*}(D_T))$. What's more, we give an explicit formula between the two index $\gamma$ and $\gamma^*$. Moreover, we prove H\"{o}lder continuity for mild solutions on bounded domains. Finally, we present a new criteria to justify H\"{o}lder continuity for the solutions on bounded domains. The novelty of this paper is that our method are suitable to the case of time-space white noise.