A central limit theorem for stochastic heat equations in random environment

TL;DR

This paper proves a central limit theorem for solutions of a one-dimensional stochastic heat equation with a random environment, showing convergence to a Gaussian distribution under diffusive scaling.

Contribution

It extends the classical CLT for finite-dimensional diffusions to an infinite-dimensional stochastic PDE setting with random nonlinear terms.

Findings

The solution's distribution converges to a Gaussian law under diffusive scaling.

The covariance operator of the limit is explicitly characterized.

The limit concentrates on constant functions in the space.

Abstract

In this article, we investigate the asymptotic behavior of the solution to a one-dimensional stochastic heat equation with random nonlinear term generated by a stationary, ergodic random field. We extend the well-known central limit theorem for finite-dimensional diffusions in random environment to this infinite-dimensional setting. Due to our result, a central limit theorem in $L^1$ sense with respect to the randomness of the environment holds under a diffusive time scaling. The limit distribution is a centered Gaussian law whose covariance operator is explicitly described. It concentrates only on the space of constant functions.