The heat equation on manifolds as a gradient flow in the Wasserstein space

We study the gradient flow for the relative entropy functional on probability measures over a Riemannian manifold. To this aim we present a notion of a Riemannian structure on the Wasserstein space. If the Ricci curvature is bounded below we establish existence and contractivity of the gradient flow...

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Bibliographic Details
Published inAnnales de l'I.H.P. Probabilités et statistiques Vol. 46; no. 1; pp. 1 - 23
Main Author Erbar, Matthias
Format Journal Article
LanguageEnglish
Published Paris Elsevier 01.02.2010
Institut Henri Poincaré
Subjects
35A15
58J35
60J60
Exact sciences and technology
General topics
Gradient flow
Heat equation
Mathematics
Probability and statistics
Probability theory and stochastic processes
Relative entropy
Sciences and techniques of general use
Statistics
Wasserstein metric
Approximation
Heat equation
35A15
60J60
Probability theory
58J35
Statistics
Probability space
Manifold
Relative entropy
Wasserstein metric
Gradient flow
Compact manifold
Probability measure
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ISSN0246-0203
DOI10.1214/08-AIHP306

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Summary:We study the gradient flow for the relative entropy functional on probability measures over a Riemannian manifold. To this aim we present a notion of a Riemannian structure on the Wasserstein space. If the Ricci curvature is bounded below we establish existence and contractivity of the gradient flow using a discrete approximation scheme. Furthermore we show that its trajectories coincide with solutions to the heat equation.
ISSN:0246-0203
DOI:10.1214/08-AIHP306