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

TL;DR

This paper establishes the spatial and temporal Hölder continuity of solutions to a semilinear stochastic fractional heat equation driven by spatially colored Gaussian noise with Riesz kernel covariance.

Contribution

It provides new regularity results for solutions to stochastic fractional heat equations with spatially colored noise, extending previous work to include Hölder continuity under these conditions.

Findings

Proves spatial Hölder continuity of solutions.

Establishes temporal Hölder continuity of solutions.

Analyzes the effects of Riesz kernel covariance on regularity.

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.