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6 - Lecture notes on gradient flows and optimal transport

from PART 1 - SHORT COURSES

Published online by Cambridge University Press:  05 August 2014

By
Sara Danieri  and
Guiseppe Savaré
Edited by
Yann Ollivier ,
Hervé Pajot  and
Cedric Villani
Sara Danieri
Affiliation:
Italy
Guiseppe Savaré
Affiliation:
Italy
Yann Ollivier
Affiliation:
Université de Paris XI
Hervé Pajot
Affiliation:
Université de Grenoble
Cedric Villani
Affiliation:
Université de Paris VI (Pierre et Marie Curie)
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Summary

Abstract

We present a short overview on the strongest variational formulation for gradient flows of geodesically λ-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School “Optimal Transportation: Theory and Applications” in Grenoble during the week of June 22–26, 2009.

Introduction

These notes are based on a series of lectures given by the second author for the Summer School “Optimal Transportation: Theory and Applications” in Grenoble during the week of June 22–26, 2009.

We try to summarize some of the main results concerning gradient flows of geodesically λ-convex functionals in metric spaces and applications to diffusion partial differential equations (PDEs) in the Wasserstein space of probability measures. Due to obvious space constraints, the theory and the references presented here are largely incomplete and should be intended as an oversimplified presentation of a quickly evolving subject. We refer to the books [3, 68] for a detailed account of the large literature available on these topics.

In the Section 6.2 we collect some elementary and well-known results concerning gradient flows of smooth convex functions in ℝd. We selected just a few topics, which are well suited for a “metric” formulation and provide a useful guide for the more abstract developments. In the Section 6.3 we present the main (and strongest) notion of gradient flow in metric spaces characterized by the solution of a metric evolution variational inequality: the aim here is to show the consequence of this definition, without any assumptions on the space and on the functional (except completeness and lower semicontinuity); we shall see that solutions to evolution variational inequalities enjoy nice stability, asymptotic, and regularization properties.

Type
Chapter
Information
Optimal Transport
Theory and Applications
 , pp. 100 - 144
Publisher: Cambridge University Press
Print publication year: 2014

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