Mesoscopic averaging of the two-dimensional KPZ equation

TL;DR

This paper investigates the mesoscopic averaging of the 2D KPZ equation, revealing a unique phenomenon where spatial averaging converges to a sum of a deterministic KPZ solution and a Gaussian variable, highlighting dimension-specific behavior.

Contribution

It introduces a novel mesoscopic averaging result for the 2D KPZ equation, demonstrating convergence to a combined deterministic and Gaussian limit, which was previously unexplored.

Findings

Averaging converges to a sum of KPZ solution and Gaussian variable

The Gaussian component depends only on the averaging scale

The phenomenon is unique to two-dimensional KPZ

Abstract

We study the limit of a local average of the KPZ equation in dimension $d=2$ with general initial data in the subcritical regime. Our result shows that a proper spatial averaging of the KPZ equation converges in distribution to the sum of the solution to a deterministic KPZ equation and a Gaussian random variable that depends solely on the scale of averaging. This shows a unique mesoscopic averaging phenomenon that is only present in dimension two. Our work is inspired by the recent findings by Chatterjee \cite{chatterjee2021weak}.