A central limit theorem for the KPZ equation

TL;DR

This paper proves a central limit theorem for the KPZ equation driven by a broad class of stationary, mixing random fields, showing convergence to the Gaussian-driven KPZ equation under suitable scaling.

Contribution

It establishes a CLT for the KPZ equation with non-Gaussian noise, extending previous results to more general random environments.

Findings

Convergence to Gaussian KPZ in the weakly asymmetric regime

Dependence of the limit on integrated variance of the noise

Higher moments influence diverging constants in scaling

Abstract

We consider the KPZ equation in one space dimension driven by a stationary centred space-time random field, which is sufficiently integrable and mixing, but not necessarily Gaussian. We show that, in the weakly asymmetric regime, the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the Hopf-Cole solution to the KPZ equation driven by space-time Gaussian white noise. While the limiting process depends only on the integrated variance of the driving field, the diverging constants appearing in the definition of the reference frame also depend on higher order moments.