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

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.