From large deviations to Wasserstein gradient flows in multiple dimensions Journal Article

Author(s): Erbar, Matthias; Maas, Jan; Renger, Michiel
Article Title: From large deviations to Wasserstein gradient flows in multiple dimensions
Affiliation IST Austria
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 Γ-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 of Duong et al 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 Adams et al to arbitrary dimensions.
Keywords: large deviations; Wasserstein metric; Gradient flows; Γ-convergence
Journal Title: Electronic Communications in Probability
Volume: 20
ISSN: 1083-589X
Publisher: Institute of Mathematical Statistics  
Date Published: 2015-11-29
Start Page: Article Number: 89
Copyright Statement: CC BY 3.0
DOI: 10.1214/ECP.v20-4315
Notes: This work has been initiated when MR visited ME and JM at the University of Bonn. The authors gratefully acknowledge support by the German Research Foundation through the Hausdorff Center for Mathematics at the University of Bonn, and the Collaborative Research Centers 1060 “The Mathematics of Emergent Effects” and 1114 “Scaling Cascades in Complex Systems”. The authors thank the referees for their useful remarks and Kaveh Bashiri for pointing out a mistake in a preliminary version of this paper.
Open access: yes (OA journal)
IST Austria Authors
  1. Jan Maas
    26 Maas
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