On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations

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