From large deviations to Wasserstein gradient flows in multiple dimensions

M. Erbar, J. Maas, M. Renger, Electronic Communications in Probability 20 (2015).

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Journal Article | Published | English

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Author
Erbar, Matthias; Maas, JanIST Austria ; Renger, Michiel
Department
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.
Publishing Year
Date Published
2015-11-29
Journal Title
Electronic Communications in Probability
Volume
20
Article Number
89
IST-REx-ID

Cite this

Erbar M, Maas J, Renger M. From large deviations to Wasserstein gradient flows in multiple dimensions. Electronic Communications in Probability. 2015;20. doi:10.1214/ECP.v20-4315
Erbar, M., Maas, J., & Renger, M. (2015). From large deviations to Wasserstein gradient flows in multiple dimensions. Electronic Communications in Probability, 20. https://doi.org/10.1214/ECP.v20-4315
Erbar, Matthias, Jan Maas, and Michiel Renger. “From Large Deviations to Wasserstein Gradient Flows in Multiple Dimensions.” Electronic Communications in Probability 20 (2015). https://doi.org/10.1214/ECP.v20-4315.
M. Erbar, J. Maas, and M. Renger, “From large deviations to Wasserstein gradient flows in multiple dimensions,” Electronic Communications in Probability, vol. 20, 2015.
Erbar M, Maas J, Renger M. 2015. From large deviations to Wasserstein gradient flows in multiple dimensions. Electronic Communications in Probability. 20.
Erbar, Matthias, et al. “From Large Deviations to Wasserstein Gradient Flows in Multiple Dimensions.” Electronic Communications in Probability, vol. 20, 89, Institute of Mathematical Statistics, 2015, doi:10.1214/ECP.v20-4315.
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2018-12-12
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