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

Abstract

Consider the following stochastic heat equation, \begin{align*} \frac{\partial u_t(x)}{\partial t}=-ν(-Δ)^{α/2} u_t(x)+σ(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 $\alpha \in (0,2]$, $\sigma :R\to R$ 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.