Fluctuations of a nonlinear stochastic heat equation in dimensions three and higher

TL;DR

This paper investigates the long-term behavior and fluctuations of solutions to a nonlinear stochastic heat equation in dimensions three and higher, driven by Gaussian noise, revealing convergence to stationarity and Edwards-Wilkinson type fluctuations.

Contribution

It establishes convergence to stationary distribution and characterizes diffusive fluctuations for the nonlinear stochastic heat equation in high dimensions.

Findings

Solution converges to stationary distribution over time

Fluctuations are described by Edwards-Wilkinson equation

Results hold for small coupling constant

Abstract

We study the solution to a nonlinear stochastic heat equation in $d\geq 3$. The equation is driven by a Gaussian multiplicative noise that is white in time and smooth in space. For a small coupling constant, we prove (i) the solution converges to the stationary distribution in large time; (ii) the diffusive scale fluctuations are described by the Edwards-Wilkinson equation.