Stochastic comparisons for stochastic heat equation

We establish the stochastic comparison principles, including moment comparison principle as a special case, for solutions to the following nonlinear stochastic heat equation on ${\mathbb{R}}^d$ $$(\frac{\mathrm{\partial }}{\mathrm{\partial }t}-\frac{1}{2}\mathrm{\Delta })u(t,x)=\rho (u(t,x))\dot{M}(t,x),$$ where $\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and colored in space, and $\rho$ is a Lipschitz continuous function that vanishes at zero. These results are obtained for rough initial data and under Dalang’s condition, namely, ${\int }_{{\mathbb{R}}^d}(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{-1}\widehat{f}(\text{d}\xi )<\mathrm{\infty }$, where $\widehat{f}$ is the spectral measure of the noise. We establish the comparison principles by comparing either the diffusion coefficient $\rho$ or the correlation function of the noise $f$. As corollaries, we obtain Slepian’s inequality for SPDEs and SDEs.