The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions

TL;DR

This paper demonstrates that a system of infinitely many weakly asymmetric interacting Brownian motions can be approximated by the KPZ equation at large scales, linking microscopic interactions to macroscopic interface fluctuations.

Contribution

It establishes the KPZ equation as the scaling limit for a specific class of weakly asymmetric interacting Brownian motions, extending the understanding of interface fluctuation models.

Findings

Large scale fluctuations are well approximated by the KPZ equation.

The proof utilizes martingale solutions and uniqueness results from prior work.

The model connects microscopic interactions to macroscopic interface behavior.

Abstract

We consider a system of infinitely many interacting Brownian motions that models the height of a one-dimensional interface between two bulk phases. We prove that the large scale fluctuations of the system are well approximated by the solution to the KPZ equation provided the microscopic interaction is weakly asymmetric. The proof is based on the martingale solutions of Goncalves and Jara and the corresponding uniqueness result of Gubinelli and Perkowski.