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

Abstract

Consider the heat equation driven by a smooth, Gaussian random potential: \begin{align*} \partial_t u_{\varepsilon}=\tfrac12Δu_{\varepsilon}+u_{\varepsilon}(ξ_{\varepsilon}-c_{\varepsilon}), \ \ t>0, x\in\mathbb{R}, \end{align*} where ${\xi }_{\epsilon }$ converges to a spacetime white noise, and $c_{\epsilon }$ is a diverging constant chosen properly. For any $n\ge 1$, we prove that $u_{\epsilon }$ 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.

15 pages, revised version

References in corpus (2)

• The Intermediate Disorder Regime for Directed Polymers in Dimension 1+1
• Diffusivity of Rescaled Random Polymer in Random Environment in dimensions 1 and 2