Open Access
2020 Stochastic comparisons for stochastic heat equation
Le Chen, Kunwoo Kim
Electron. J. Probab. 25: 1-38 (2020). DOI: 10.1214/20-EJP541


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.


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Le Chen. Kunwoo Kim. "Stochastic comparisons for stochastic heat equation." Electron. J. Probab. 25 1 - 38, 2020.


Received: 17 March 2020; Accepted: 22 October 2020; Published: 2020
First available in Project Euclid: 10 December 2020

MathSciNet: MR4186259
Digital Object Identifier: 10.1214/20-EJP541

Primary: 60H15
Secondary: 35R60 , 60G60

Keywords: infinite dimensional SDE , moment comparison principle , Parabolic Anderson model , rough initial data , Slepian’s inequality for SPDEs , spatially homogeneous noise , stochastic comparison principle , Stochastic heat equation

Vol.25 • 2020
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