Stochastic comparisons for stochastic heat equation

Le Chen,Kunwoo Kim

doi:10.1214/20-ejp541

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.

Highlights

We study the stochastic comparison principle including moment comparison principle for the solutions to the following stochastic heat
For some set of functions F, such as those defined in Definition 1.2, and for some n ≥ 1, we say that u1 and u2 satisfy the n-time stochastic comparison principle over F with u1 dominating u2 if for any 0 < t1 < · · · < tn < ∞, and F1, . . . , Fn ∈ F, it holds that n n
We show that stochastic heat equations on Rd with rough initial condition and driven by Gaussian noise which is white in time and correlated in space can be approximated by systems of interacting diffusions on the d-dimensional lattice

Summary

Introduction

We study the stochastic comparison principle (see Definition 1.4) including moment comparison principle for the solutions to the following stochastic heat In this equation, ρ is assumed to be a globally Lipschitz continuous function with ρ(0) = 0. When the noise is additive, i.e., ρ(u) = constant, the moment comparison principle — Case (E-1) — under the second scenario (S-2) comes from Isserlis’ theorem [19] since the solution is a Gaussian random field whose distribution is determined by the spatial correlation function f. In both [20] and [13], the initial conditions are assumed to be the Lebesgue measure μ(dx) = dx We will generalize these results to cover rough initial data and all possible correlation functions under Dalang’s condition (1.6). We believe strong solutions are more straightforward and easier to handle when showing approximations

Statement of the main results

Outline of the paper

Stochastic comparison principles for interacting diffusions

Several approximations

Comparison theorems for finite interacting diffusions

Some examples and one application