Gaussian fluctuations of a nonlinear stochastic heat equation in dimension two

Abstract

We study the Gaussian fluctuations of a nonlinear stochastic heat equation in spatial dimension two. The equation is driven by a Gaussian multiplicative noise. The noise is white in time, smoothed in space at scale \(\varepsilon \), and tuned logarithmically by a factor \(\frac{1}{\sqrt{\log \varepsilon ^{-1}}}\) in its strength. We prove that, after centering and rescaling, the solution random field converges in distribution to an Edwards-Wilkinson limit as \(\varepsilon \downarrow 0\). The tool we used here is the Malliavin-Stein’s method. We also give a functional version of this result.