Convergence of the KMP model to the KPZ equation

TL;DR

This paper proves that the KMP process converges to the KPZ equation over large times, using a novel approach involving stochastic flows and recent convergence results for random walks in random environments.

Contribution

It establishes the convergence of the KMP process to the KPZ equation by connecting it with stochastic flows and recent theoretical results.

Findings

KMP process converges to KPZ in a scaled window

Identification of KMP with a stochastic flow of kernels

Application of recent random walk convergence results

Abstract

We prove that the Kipnis-Marchioro-Presutti (KMP) process converges to the Kardar-Parisi-Zhang (KPZ) equation, as time $t$ goes to infinity, in a properly scaled observation window shifted by $t^{3/4}$. Our proof is based on identifying the KMP process with a stochastic flow of kernels describing transition probabilities in a certain model of random walk in space-time random environment. This allows to apply a recent result of arXiv:2401.06073 proving convergence of the density field of random walks in random environment to the KPZ equation in a suitably general sense.