H$\ddot{o}$lder continuity for stochastic fractional heat equation with colored noise

Abstract

In this paper, we consider semilinear stochastic fractional heat equation $\frac{\partial }{\partial t}{u}_{\beta ,t}(x)={△}^{\alpha /2}{u}_{\beta ,t}(x)+\sigma (u_{\beta ,t}(x)){\eta }_{\beta }$. The Gaussian noise ${\eta }_{\beta }$ is assumed to be colored in space with covariance of the form $E({\eta }_{\beta }(t,x){\eta }_{\beta }(s,y))=\delta (t-s)f_{\beta }(x-y)$, where $f_{\beta }$ is the Riesz kernel $f_{\beta }(x)\propto |x{|}^{-\beta }$. We obtain the spatial and temporal H$\ddot{\mbox{o}}$lder continuity of the mild solution.