Wasserstein gradient flows from large deviations of thermodynamic limits

TL;DR

This paper links large deviations of stochastic particle systems to Wasserstein gradient flows, providing a new formulation of the rate functional that explicitly involves free energy and establishing asymptotic equivalence with gradient discretization schemes.

Contribution

It introduces a novel formulation of the large deviation rate functional that explicitly involves free energy and connects it to Wasserstein gradient flows through Gamma-convergence.

Findings

Explicit rate functional involving free energy

Asymptotic Gamma-convergence of discrete rate functional

Connection between large deviations and Wasserstein gradient flows

Abstract

We study the Fokker-Planck equation as the hydrodynamic limit of a stochastic particle system on one hand and as a Wasserstein gradient flow on the other. We write the rate functional, that characterizes the large deviations from the hydrodynamic limit, in a way that the free energy appears explicitly. Next we use this formulation via the contraction principle to prove that the discreet time rate functional is asymptotically equivalent in the Gamma-convergence sense to the functional derived from the Wasserstein gradient discretization scheme.