Another look into the Wong-Zakai Theorem for Stochastic Heat Equation

TL;DR

This paper revisits the Wong-Zakai theorem for the stochastic heat equation, demonstrating convergence of solutions driven by smooth Gaussian potentials to the SPDE solution, offering a probabilistic perspective on known results.

Contribution

It provides a probabilistic proof of convergence for solutions driven by smooth approximations to white noise, specifically for the stochastic heat equation, complementing existing analytical approaches.

Findings

$u_ ext{ ext{}} o$ converges in $L^n$ to the SPDE solution

Probabilistic approach offers new perspective on Wong-Zakai theorem

Discussion on transition from homogenization to stochasticity

Abstract

Consider the heat equation driven by a smooth, Gaussian random potential: \begin{align*} \partial_t u_{\varepsilon}=\tfrac12\Delta u_{\varepsilon}+u_{\varepsilon}(\xi_{\varepsilon}-c_{\varepsilon}), \ \ t>0, x\in\mathbb{R}, \end{align*} where $\xi_{\varepsilon}$ converges to a spacetime white noise, and $c_{\varepsilon} $ is a diverging constant chosen properly. For any $ n\geq 1 $, we prove that $ u_{\varepsilon} $ converges in $ L^n $ to the solution of the stochastic heat equation. Our proof is probabilistic, hence provides another perspective of the general result of Hairer and Pardoux \cite{Hairer15a}, for the special case of the stochastic heat equation. We also discuss the transition from homogenization to stochasticity.