Conservative stochastic PDE and fluctuations of the symmetric simple exclusion process

TL;DR

This paper introduces a stochastic PDE model with conservative noise to accurately describe and analyze the fluctuations and rare events of the symmetric simple exclusion process around its hydrodynamic limit.

Contribution

It develops a continuum stochastic PDE framework that captures both typical fluctuations and rare deviations of the exclusion process, linking PDE solutions to particle system behavior.

Findings

SPDE solutions match particle fluctuations to first order

Model accurately captures rare events in particle system

Zero-noise large deviations principle established

Abstract

In this paper, we provide a continuum model for the fluctuations of the symmetric simple exclusion process about its hydrodynamic limit. The model is based on an approximating sequence of stochastic PDEs with nonlinear, conservative noise. In the small-noise limit, we show that the fluctuations of the solutions are to first-order the same as the fluctuations of the particle system. Furthermore, the SPDEs correctly simulate the rare events in the particle process. We prove that the solutions satisfy a zero-noise large deviations principle with rate function equal to the rate function describing the deviations of the symmetric simple exclusion process from its hydrodynamic limit.