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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^d $\end{document}</tex-math></inline-formula>) for mild solutions of stochastic nonlocal diffusion equations in the sense that the solutions belong to the space <inline-formula><tex-math id="M2">\begin{document}$ C^{\gamma}(D_T;L^p(\Omega)) $\end{document}</tex-math></inline-formula> with the optimal Hölder continuity index <inline-formula><tex-math id="M3">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> (which is given explicitly), where <inline-formula><tex-math id="M4">\begin{document}$ D_T: = [0, T]\times D $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ T&gt;0 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M6">\begin{document}$ D\subset\mathbb{R}^d $\end{document}</tex-math></inline-formula> being a bounded domain. Then, by utilising tail estimates, we are able to obtain the estimates of mild solutions in <inline-formula><tex-math id="M7">\begin{document}$ L^p(\Omega;C^{\gamma^*}(D_T)) $\end{document}</tex-math></inline-formula>. What's more, we give an explicit formula between the two indexes <inline-formula><tex-math id="M8">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \gamma^* $\end{document}</tex-math></inline-formula>. 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.</p>