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 $\epsilon$, and tuned logarithmically by a factor $\frac{1}{\sqrt{\log {\epsilon }^{-1}}}$ in its strength. We prove that, after centering and rescaling, the solution random field converges in distribution to an Edwards-Wilkinson limit as $\epsilon \downarrow 0$. The tool we used here is the Malliavin-Stein's method. We also give a functional version of this result.

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