Stochastic Burgers Equation via Energy Solutions from Non-Stationary Particle Systems

Abstract

We prove that the stochastic Burgers equation, which is related to the Kardar-Parisi-Zhang/KPZ equation via weak derivative, is a "critical" scaling limit for density fluctuations for a family of non-integrable and non-stationary interacting particle systems. The models we consider cannot be linearized by a microscopic Cole-Hopf transform or studied directly by the existing energy solution theory of Goncalves-Jara '14. We develop a novel method based on comparison to stationary models, a technique that has not yet been applied to universality of the KPZ equation and nontrivially expands the set of models for which universality is confirmed. We also study crossover fluctuations and prove, for the first time, the full transition and phase diagram from Gaussian to KPZ fluctuations for non-stationary interacting particle systems, which has not even been done yet for integrable models. We emphasize the method developed herein applies to a general class of models/particle systems, but we restrict to a class of zero-range systems whose non-stationary versions have received widespread interest but had not been treated in the context of KPZ until now as well as class of non-simple exclusion processes that we comment on.