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Wasserstein gradient flows from large deviations ofmany-particle limits

Published online by Cambridge University Press:  13 August 2013

Manh Hong Duong ,
Vaios Laschos  and
Michiel Renger
Manh Hong Duong
Affiliation:
Department of Mathematical sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom
Vaios Laschos
Affiliation:
Department of Mathematical sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom
Michiel Renger
Affiliation:
ICMS and Dep. of Math. and Comp. Sciences, TU Eindhoven, Den Dolech 2, 5612 AZ Eindhoven, the Netherlands. d.r.m.renger@tue.nl

Abstract

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We study the Fokker–Planck equation as the many-particle limit of a stochastic particlesystem on one hand and as a Wasserstein gradient flow on the other. We write thepath-space rate functional, which characterises the large deviations from the expectedtrajectories, in such a way that the free energy appears explicitly. Next we use thisformulation via the contraction principle to prove that the discrete time rate functionalis asymptotically equivalent in the Gamma-convergence sense to the functional derived fromthe Wasserstein gradient discretization scheme.

Keywords

Wassersteingradient flowsFokker–PlanckGamma-convergencelarge deviations

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 4 , October 2013 , pp. 1166 - 1188
Copyright
© EDP Sciences, SMAI, 2013

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