Gaussian fluctuations of a nonlinear stochastic heat equation in dimension two

TL;DR

This paper investigates the Gaussian fluctuations of a nonlinear stochastic heat equation in two dimensions driven by a specific Gaussian noise, demonstrating convergence to an Edwards-Wilkinson limit using Malliavin-Stein's method.

Contribution

It establishes the Gaussian fluctuation limit for a nonlinear stochastic heat equation in 2D with a novel noise scaling, extending the understanding of such equations.

Findings

Convergence to Edwards-Wilkinson limit after rescaling.

Use of Malliavin-Stein's method for the proof.

Functional version of the fluctuation result.

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.