Spatial asymptotics and strong comparison principle for some fractional stochastic heat equations

TL;DR

This paper investigates the spatial asymptotic behavior and comparison principles for solutions to fractional stochastic heat equations driven by Gaussian noise, extending recent theoretical results in the field.

Contribution

It establishes a strong comparison theorem and analyzes how initial data influence the spatial asymptotics of solutions to fractional stochastic heat equations.

Findings

Proved a strong comparison principle for the solutions.

Analyzed the impact of initial data on spatial asymptotics.

Extended existing results to fractional Laplacian cases.

Abstract

Consider the following stochastic heat equation, \begin{align*} \frac{\partial u_t(x)}{\partial t}=-\nu(-\Delta)^{\alpha/2} u_t(x)+\sigma(u_t(x))\dot{F}(t,\,x), \quad t>0, \; x \in R^d. \end{align*} Here $-\nu(-\Delta)^{\alpha/2}$ is the fractional Laplacian with $\nu>0$ and $\alpha \in (0,2]$, $\sigma: R\rightarrow R$ is a globally Lipschitz function, and $\dot{F}(t,\,x)$ is a Gaussian noise which is white in time and colored in space. Under some suitable additional conditions, we prove a strong comparison theorem and explore the effect of the initial data on the spatial asymptotic properties of the solution. This constitutes an important extension over a series of recent works.