From large deviations to Wasserstein gradient flows in multiple dimensions

TL;DR

This paper extends the understanding of large deviation principles for Brownian particles to multiple dimensions, establishing a key upper bound in the Wasserstein gradient flow framework beyond one dimension.

Contribution

It provides a new proof of the upper bound for the large deviation rate functional in higher dimensions, generalizing previous one-dimensional results.

Findings

Established the upper bound in the large deviation principle for multiple dimensions.

Connected large deviations with Wasserstein gradient flows in higher dimensions.

Extended prior one-dimensional results to arbitrary dimensions.

Abstract

We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer that this functional is asymptotically equivalent (in the sense of $\Gamma$-convergence) to the Jordan--Kinderlehrer--Otto functional arising in the Wasserstein gradient flow structure of the Fokker--Planck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in \cite{DLR2013} relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of \cite{ADPZ2011} to arbitrary dimensions.