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

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 $ρ$ is Lipschitz continuous. These results are obtained under the condition that $\int_{\mathbb{R}^d}(1+|ξ|^2)^{α-1}\hat{f}(\text{d} ξ)<\infty$ for some $α\in(0,1]$, 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., $α=0$.} As some intermediate results, we obtain handy upper bounds for $L^p(Ω)$-moments of $u(t,x)$ for all $p\ge 2$, and also prove that $u$ is a.s. Hölder continuous with order $α-ε$ in space and $α/2-ε$ in time for any small $ε>0$.

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