Another Look into the Wong–zakai Theorem for Stochastic Heat Equation

Consider the heat equation driven by a smooth, Gaussian random potential: ∂tuε = 1 2 ∆uε + uε(ξε − cε), t > 0, x ∈ R, where ξε converges to a spacetime white noise, and cε is a diverging constant chosen properly. For any n > 1, we prove that uε converges in Ln to the solution of the stochastic heat equation. Our proof is probabilistic, hence provides another perspective of the general result of Hairer and Pardoux [HP15], for the special case of the stochastic heat equation.