Gamma Convergence Approach For The Large Deviations Of The Density In Systems Of Interacting Diffusion Processes

TL;DR

This paper investigates the large deviations of the empirical density in systems of interacting diffusions, using Gamma convergence to connect microscopic large deviation principles with macroscopic fluctuation theory.

Contribution

It establishes the Gamma convergence of the rate functionals for the empirical density as the scaling parameter tends to zero, linking microscopic and macroscopic descriptions.

Findings

Gamma convergence of rate functionals is proven as the scale parameter approaches zero.

The limiting functional aligns with the one in Macroscopic Fluctuations Theory.

Provides a rigorous connection between microscopic large deviations and macroscopic fluctuation descriptions.

Abstract

We consider extended slow-fast systems of N interacting diffusions. The typical behavior of the empirical density is described by a nonlinear McKean-Vlasov equation depending on , the scaling parameter separating the time scale of the slow variable from the time scale of the fast variable. Its atypical behavior is encapsulated in a large N Large Deviation Principle (LDP) with a rate functional. We study the $\Gamma$-convergence of as $\rightarrow$ 0 and show it converges to the rate functional appearing in the Macroscopic Fluctuations Theory (MFT) for diffusive systems.