KPZ equation from a class of nonlinear SPDEs in infinite volume

TL;DR

This paper rigorously derives the KPZ equation as a continuum limit from a broad class of nonlinear stochastic PDEs in infinite volume, confirming its physical origins in a full-space setting.

Contribution

It provides a rigorous mathematical derivation of the KPZ equation from nonlinear SPDEs in infinite volume, addressing a longstanding open problem.

Findings

KPZ equation derived as a limit of nonlinear SPDEs

Use of stochastic heat kernel for linearized equations

Addresses full-space derivation problem posed by Hairer-Quastel

Abstract

We study a general class of nonlinear Ginzburg-Landau SPDEs in infinite volume under weak nonlinearity scaling and with non-equilibrium initial data. We derive the KPZ equation as a continuum limit of these equations. This makes rigorous the original derivation of the KPZ equation from physics in the full-space setting, which was a problem posed by Hairer-Quastel '18. Our analysis is based on a stochastic heat kernel for a linearization of said SPDEs.