Comparison principle for stochastic heat equation on ${\mathbb{R}}^d$

Abstract

We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on ${\mathbb{R}}^d$ \[ \left(\frac{\partial }{\partial t} -\frac{1}{2}Δ\right) u(t,x) = ρ(u(t,x)) \:\dot{M}(t,x), \] for measure-valued initial data, where $\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and $\ddot{I}$ is Lipschitz continuous. These results are obtained under the condition that ${\int }_{{\mathbb{R}}^d}(1+|\hat{I}¾{|}^2{)}^{\hat{I}\pm -1}\hat{f}(\text{d}\hat{I}¾)<\infty$ for some $\hat{I}\pm \in (0,1]$, where $\hat{f}$ is the spectral measure of the noise. {The weak comparison principle and nonnegativity of solutions to the same equation are obtained under Dalang's condition, i.e., $\hat{I}\pm =0$.} As some intermediate results, we obtain handy upper bounds for $L^p(\hat{I}\text{c◯})$-moments of $u(t,x)$ for all $p\ge 2$, and also prove that $u$ is a.s. Hölder continuous with order $\hat{I}\pm -\hat{I}\mu$ in space and $\hat{I}\pm /2-\hat{I}\mu$ in time for any small $\hat{I}\mu >0$.