Concurrent Donsker-Varadhan and hydrodynamical large deviations

TL;DR

This paper establishes a large deviations principle for the empirical density and current in a weakly asymmetric exclusion process on a torus, linking variational convergence and hydrodynamical rate functions to analyze rare events.

Contribution

It introduces a joint large deviations framework for empirical density and current, combining variational analysis and hydrodynamical methods for the first time in this context.

Findings

Large deviations principle for empirical density and current.

Identification of dynamical phase transitions via minimizer structure.

Connection between variational convergence and hydrodynamical rate functions.

Abstract

We consider the weakly asymmetric exclusion process on the $d$-dimensional torus. We prove a large deviations principle for the time averaged empirical density and current in the joint limit in which both the time interval and the number of particles diverge. This result is obtained both by analyzing the variational convergence, as the number of particles diverges, of the Donsker-Varadhan functional for the empirical process and by considering the large time behavior of the hydrodynamical rate function. The large deviations asymptotic of the time averaged current is then deduced by contraction principle. The structure of the minimizers of this variational problem corresponds to the possible occurrence of dynamical phase transitions.