Regularity of stochastic nonlocal diffusion equations

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 Hölder 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 Hölder 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 $L^p(\Omega ;C^{{\gamma }^{\ast }}(D_T))$. What's more, we give an explicit formula between the two index $\gamma$ and ${\gamma }^{\ast }$. Moreover, we prove Hölder continuity for mild solutions on bounded domains. Finally, we present a new criteria to justify Hölder 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.

23 pages