The Annals of Probability

Comparison principle for stochastic heat equation on $\mathbb{R}^{d}$

Le Chen and Jingyu Huang

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We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on $\mathbb{R}^{d}$

\[\biggl(\frac{\partial}{\partial t}-\frac{1}{2}\Delta \biggr)u(t,x)=\rho\bigl(u(t,x)\bigr)\dot{M}(t,x),\] for measure-valued initial data, where $\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and $\rho$ is Lipschitz continuous. These results are obtained under the condition that $\int_{\mathbb{R}^{d}}(1+|\xi|^{2})^{\alpha-1}\hat{f}(\text{d}\xi)<\infty$ for some $\alpha\in(0,1]$, where $\hat{f}$ is the spectral measure of the noise. The weak comparison principle and nonnegativity of solutions to the same equation are obtained under Dalang’s condition, that is, $\alpha=0$. As some intermediate results, we obtain handy upper bounds for $L^{p}(\Omega)$-moments of $u(t,x)$ for all $p\ge2$, and also prove that $u$ is a.s. Hölder continuous with order $\alpha-\varepsilon$ in space and $\alpha/2-\varepsilon$ in time for any small $\varepsilon>0$.

Article information

Ann. Probab., Volume 47, Number 2 (2019), 989-1035.

Received: July 2016
Revised: November 2017
First available in Project Euclid: 26 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 60G60: Random fields

Stochastic heat equation parabolic Anderson model space-time Hölder regularity spatially homogeneous noise comparison principle measure-valued initial data


Chen, Le; Huang, Jingyu. Comparison principle for stochastic heat equation on $\mathbb{R}^{d}$. Ann. Probab. 47 (2019), no. 2, 989--1035. doi:10.1214/18-AOP1277.

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