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October 2019 Another look into the Wong–Zakai theorem for stochastic heat equation
Yu Gu, Li-Cheng Tsai
Ann. Appl. Probab. 29(5): 3037-3061 (October 2019). DOI: 10.1214/19-AAP1474

Abstract

For the heat equation driven by a smooth, Gaussian random potential: \begin{equation*}\partial_{t}u_{\varepsilon}=\frac{1}{2}\Delta u_{\varepsilon}+u_{\varepsilon}(\xi_{\varepsilon}-c_{\varepsilon}),\quad t>0,x\in\mathbb{R},\end{equation*} where $\xi_{\varepsilon}$ converges to a spacetime white noise, and $c_{\varepsilon}$ is a diverging constant chosen properly, we prove that $u_{\varepsilon}$ converges in $L^{n}$ to the solution of the stochastic heat equation for any $n\geq1$. Our proof is probabilistic, hence provides another perspective of the general result of Hairer and Pardoux (J. Math. Soc. Japan 67 (2015) 1551–1604), for the special case of the stochastic heat equation. We also discuss the transition from homogenization to stochasticity.

Citation

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Yu Gu. Li-Cheng Tsai. "Another look into the Wong–Zakai theorem for stochastic heat equation." Ann. Appl. Probab. 29 (5) 3037 - 3061, October 2019. https://doi.org/10.1214/19-AAP1474

Information

Received: 1 February 2018; Revised: 1 October 2018; Published: October 2019
First available in Project Euclid: 18 October 2019

zbMATH: 07155066
MathSciNet: MR4019882
Digital Object Identifier: 10.1214/19-AAP1474

Subjects:
Primary: 35R60, 60H15
Secondary: 60H07

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 5 • October 2019
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