Stochastic comparisons for stochastic heat equation

Abstract

We establish the stochastic comparison principles, including moment comparison principle as a special case, for solutions to the following nonlinear stochastic heat equation on ${\mathbb{R}}^d$ \[ \left(\frac{\partial }{\partial t} -\frac{1}{2}Δ\right) u(t,x) = ρ(u(t,x)) \:\dot{M}(t,x), \] 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, ${\int }_{{\mathbb{R}}^d}(1+|\xi {|}^2{)}^{-1}\hat{f}(\text{d}\xi )<\infty$, where $\hat{f}$ 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.

38 pages, 0 figure