The Annals of Applied Probability

Another look into the Wong–Zakai theorem for stochastic heat equation

Yu Gu and Li-Cheng Tsai

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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.

Article information

Ann. Appl. Probab., Volume 29, Number 5 (2019), 3037-3061.

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

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Digital Object Identifier

Primary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus

Stochastic heat equation Feynman–Kac formula Wiener chaos expansion


Gu, Yu; Tsai, Li-Cheng. Another look into the Wong–Zakai theorem for stochastic heat equation. Ann. Appl. Probab. 29 (2019), no. 5, 3037--3061. doi:10.1214/19-AAP1474.

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