Local Brownian property of the narrow wedge solution of the KPZ equation

TL;DR

This paper proves that the narrow wedge solution of the KPZ equation exhibits local Brownian properties and remains within certain initial conditions over time, using a simplified proof approach.

Contribution

It provides a simple proof that the KPZ solution with narrow wedge initial data has local Brownian properties and persists within initial condition classes.

Findings

H(t,x) - (H(t,x) - H^{eq}(t,0)) is locally of finite variation

Solutions starting with Brownian motion plus Lipschitz functions stay within that class

Simplified proof technique for local Brownian property of KPZ solutions

Abstract

Let H(t,x) be the Hopf-Cole solution at time t of the Kardar-Parisi-Zhang (KPZ) equation starting with narrow wedge initial condition, i.e. the logarithm of the solution of the multiplicative stochastic heat equation starting from a Dirac delta. Also let H^{eq}(t,x) be the solution at time t of the KPZ equation with the same noise, but with initial condition given by a standard two-sided Brownian motion, so that H^{eq}(t,x)-H^{eq}(0,x) is itself distributed as a standard two-sided Brownian motion. We provide a simple proof of the following fact: for fixed t, H(t,x)-(H(t,x)-H^{eq}(t,0)) is locally of finite variation. Using the same ideas we also show that if the KPZ equation is started with a two-sided Brownian motion plus a Lipschitz function then the solution stays in this class for all time.