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

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.