Macroscopic fluctuation theory from a Lagrangian viewpoint and the Schr\"odinger problem

TL;DR

This paper extends the Schr"odinger problem to interacting particle systems in the hydrodynamical regime, analyzing large deviations of empirical measures and currents, and characterizes optimal measures under various constraints.

Contribution

It introduces a formulation of the Schr"odinger problem for interacting particles, expanding the standard independent particle framework and analyzing constraints related to density and current.

Findings

Characterization of optimal measures for initial and final density constraints

Extension of the Schr"odinger problem to current-related constraints

Analysis of large deviations rate functions for empirical measures

Abstract

We formulate the Schr\"odinger problem for interacting particle systems in the hydrodynamical regime thus extending the standard setting of independent particles. This involves the large deviations rate function for the empirical measure which is in fact a richer observable than the hydrodynamic observables density and current. In the case in which the constraints are the initial and final density, we characterize the optimal measure for the Schr\"odinger problem. We also introduce versions of the Schr\"odinger problem in which the constraints are related to the current and analyze the corresponding optimal measures.