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March 2019 Comparison principle for stochastic heat equation on $\mathbb{R}^{d}$
Le Chen, Jingyu Huang
Ann. Probab. 47(2): 989-1035 (March 2019). DOI: 10.1214/18-AOP1277

Abstract

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 $\varepsilon>0$.

Citation

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Le Chen. Jingyu Huang. "Comparison principle for stochastic heat equation on $\mathbb{R}^{d}$." Ann. Probab. 47 (2) 989 - 1035, March 2019. https://doi.org/10.1214/18-AOP1277

Information

Received: 1 July 2016; Revised: 1 November 2017; Published: March 2019
First available in Project Euclid: 26 February 2019

zbMATH: 07053562
MathSciNet: MR3916940
Digital Object Identifier: 10.1214/18-AOP1277

Subjects:
Primary: 60H15
Secondary: 35R60, 60G60

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 2 • March 2019
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