## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 6B (2017), 4167-4221.

### A central limit theorem for the KPZ equation

#### 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.

#### Article information

**Source**

Ann. Probab. Volume 45, Number 6B (2017), 4167-4221.

**Dates**

Received: July 2015

Revised: August 2016

First available in Project Euclid: 12 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1513069258

**Digital Object Identifier**

doi:10.1214/16-AOP1162

**Subjects**

Primary: 60H15: Stochastic partial differential equations [See also 35R60]

Secondary: 35K55: Nonlinear parabolic equations 60H30: Applications of stochastic analysis (to PDE, etc.)

**Keywords**

KPZ equation central limit theorem Wiener chaos cumulants

#### Citation

Hairer, Martin; Shen, Hao. A central limit theorem for the KPZ equation. Ann. Probab. 45 (2017), no. 6B, 4167--4221. doi:10.1214/16-AOP1162. https://projecteuclid.org/euclid.aop/1513069258