An invariance principle for the 1D KPZ equation

TL;DR

This paper proves an invariance principle showing that certain discrete random surface growth models converge to the continuous 1D KPZ equation's solution under specific conditions, extending classical invariance results to nonlinear stochastic PDEs.

Contribution

It establishes a convergence result for a class of discrete surface growth models to the KPZ equation, providing a rigorous link between discrete and continuous stochastic processes.

Findings

Discrete models converge to KPZ solution as noise variance vanishes

Conditions include shift-equivariance, symmetry, and smoothness of the growth function

Provides a nonlinear invariance principle analogous to Donsker's theorem

Abstract

Consider a discrete one-dimensional random surface whose height at a point grows as a function of the heights at neighboring points plus an independent random noise. Assuming that this function is equivariant under constant shifts, symmetric in its arguments, and at least six times continuously differentiable in a neighborhood of the origin, we show that as the variance of the noise goes to zero, any such process converges to the Cole-Hopf solution of the 1D KPZ equation under a suitable scaling of space and time. This proves an invariance principle for the 1D KPZ equation, in the spirit of Donsker's invariance principle for Brownian motion.