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2015 From large deviations to Wasserstein gradient flows in multiple dimensions
Matthias Erbar, Jan Maas, Michiel Renger
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Electron. Commun. Probab. 20: 1-12 (2015). DOI: 10.1214/ECP.v20-4315


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 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.


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Matthias Erbar. Jan Maas. Michiel Renger. "From large deviations to Wasserstein gradient flows in multiple dimensions." Electron. Commun. Probab. 20 1 - 12, 2015.


Accepted: 29 November 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1333.35294
MathSciNet: MR3434206
Digital Object Identifier: 10.1214/ECP.v20-4315

Primary: 35A15
Secondary: 35Q84 , 46N55 , 60F10

Keywords: gradient flows , large deviations , Wasserstein metric , Γ-convergence

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