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 $\mathbb{R} ^{d}$ \[ \left (\frac{\partial } {\partial t} -\frac{1} {2}\Delta \right ) u(t,x) = \rho (u(t,x)) \:\dot{M} (t,x), \] 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+|\xi |^{2})^{-1}\hat{f} (\text{d} \xi )<\infty $, where $\hat{f} $ is the spectral measure of the noise. We first show that the nonlinear stochastic heat equation can be approximated by systems of interacting diffusions (SDEs) and then, using those approximations, 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.