The Annals of Probability

A central limit theorem for the KPZ equation

Martin Hairer and Hao Shen

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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

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

Received: July 2015
Revised: August 2016
First available in Project Euclid: 12 December 2017

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Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35K55: Nonlinear parabolic equations 60H30: Applications of stochastic analysis (to PDE, etc.)

KPZ equation central limit theorem Wiener chaos cumulants


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.

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