An invariance principle for stochastic heat equations with periodic coefficients

TL;DR

This paper establishes an invariance principle for the stochastic heat equation with periodic coefficients, showing weak convergence to a Gaussian process and extending the central limit theorem to infinite dimensions.

Contribution

It extends the central limit theorem to infinite-dimensional stochastic heat equations with periodic nonlinearities, providing an invariance principle and detailed covariance characterization.

Findings

Weak convergence of scaled solutions to Gaussian variables

Vanishing spatial fluctuation in the limit

Invariance principle for rescaled solutions as epsilon approaches zero

Abstract

We investigate the asymptotic behaviors of the solution $u(t, \cdot)$ to a stochastic heat equation with a periodic, gradient-type nonlinear term. We extend the central limit theorem for finite-dimensional diffusions to infinite-dimensional settings. Due to our results, $\frac{1}{\sqrt t}u(t, \cdot)$ converges weakly to a centered Gaussian variable whose covariance operator is described through Poisson equations. Different from the finite-dimensional case, the fluctuation in space vanishes in the limit distribution. Furthermore, we verify the tightness and present an invariance principle for $\{\epsilon u(\epsilon^{-2}t, \cdot)\}_{t \in [0, T]}$ as $\epsilon \downarrow 0$.