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

Abstract: In this paper, we consider semilinear stochastic fractional heat equation $\frac{\partial}{\partial t}u_{\beta,t}(x)=\triangle^{\alpha/2}u_{\beta,t}(x)+\sigma(u_{\beta,t}(x))\eta_{\beta}$. 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.