Spatial decorrelation of KPZ from narrow wedge

TL;DR

This paper investigates the spatial decorrelation properties of the KPZ equation with narrow wedge initial data, revealing decay rates of covariance and convergence to Brownian motion for spatial averages.

Contribution

It provides the first precise decay rate of spatial covariance and proves convergence of spatial averages to Brownian motion for the KPZ equation.

Findings

Covariance decays as t/x for large x

Spatial averages converge to Brownian motion

Quantitative understanding of spatial decorrelation in KPZ

Abstract

We study the spatial decorrelation of the solution to the KPZ equation with narrow wedge initial data. For fixed $t>0$, we determine the decay rate of the spatial covariance function, showing that ${\rm Cov}[h(t,x),h(t,0)]\sim \frac{t}{x}$ as $x\to\infty$. In addition, we prove that the finite-dimensional distributions of the properly rescaled spatial average of the height function converge to those of a Brownian motion.