Stochastic comparisons for stochastic heat equation

TL;DR

This paper develops stochastic comparison principles for solutions to nonlinear stochastic heat equations driven by Gaussian noise, extending existing inequalities and accommodating rough initial data under Dalang's condition.

Contribution

It introduces new stochastic comparison principles for SPDEs with rough initial data, comparing diffusion coefficients and noise correlation functions, and derives Slepian's inequalities.

Findings

Established moment comparison principles for SPDE solutions.

Derived Slepian's inequality for SPDEs and SDEs.

Extended comparison results to rough initial data under Dalang's condition.

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 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.