Finsler structure in the <i>p</i>-Wasserstein space and gradient flows

Martial Agueh

doi:10.1016/j.crma.2011.11.014

Partial Differential Equations/Differential Geometry

Finsler structure in the p-Wasserstein space and gradient flows
[Structure de Finsler dans lʼespace de Wasserstein

L^{p}

et flux de gradient]

Présenté par : Gilles Pisier

Martial Agueh ¹

¹ Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, BC, V8W 3R4, Canada

Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 35-40.

Résumé
Abstract

Il est connu que lʼespace des mesures de probabilités muni de la distance de Wasserstein $L^{2}$ (lʼespace de Wasserstein $L^{2}$ ) est une variété Riemanienne (voir F. Otto (2001) [9]). Ici, nous montrons que lorsquʼon change le coût quadratique en un coût plus general, homogène de degré $p > 1$ , lʼespace correspondant (lʼespace de Wasserstein $L^{p}$ ) admet une structure de Finsler dont la distance induite est la distance de Wasserstein $L^{p}$ . Grâce à cette structure de Finsler, nous donnons une définition de la différentiel et du gradient des fonctionelles définies sur cet espace, et aussi des flux de gradient sur cet espace. En particulier nous montrons que lʼéquation parabolique q-Laplacien est un flux de gradient dans lʼespace de Wasserstein $L^{p}$ pour $p = q / (q - 1)$ . Quand $p = 2$ , nous retrouvons la structure Remannienne de F. Otto, ce qui confirme que lʼespace de Wasserstein $L^{2}$ est une variété Riemanienne de Finsler. Notre méthode sʼapplique à des mesures de probabilité absolument continues par rapport à la mesure de Lebesgue dans $R^{n}$ , et dont les densités sont strictement positives.

It is known from the work of F. Otto (2001) [9], that the space of probability measures equipped with the quadratic Wasserstein distance, i.e., the 2-Wasserstein space, can be viewed as a Riemannian manifold. Here we show that when the quadratic cost is replaced by a general homogeneous cost of degree $p > 1$ , the corresponding space of probability measures, i.e., the p-Wasserstein space, can be endowed with a Finsler metric whose induced distance function is the p-Wasserstein distance. Using this Finsler structure of the p-Wasserstein space, we give definitions of the differential and gradient of functionals defined on this space, and then of gradient flows in this space. In particular we show in this framework that the parabolic q-Laplacian equation is a gradient flow in the p-Wasserstein space, where $p = q / (q - 1)$ . When $p = 2$ , we recover the Riemannian structure introduced by F. Otto, which confirms that the 2-Wasserstein space is a Riemann–Finsler manifold. Our approach is confined to a smooth situation where probability measures are absolutely continuous with respect to the Lebesgue measure on $R^{n}$ , and they have smooth and strictly positive densities.

Reçu le : 2011-02-17
Accepté le : 2011-11-24
Publié le : 2011-12-08

DOI : 10.1016/j.crma.2011.11.014

Affiliations des auteurs :

Martial Agueh ¹

¹ Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, BC, V8W 3R4, Canada

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     author = {Martial Agueh},
     title = {Finsler structure in the {\protect\emph{p}-Wasserstein} space and gradient flows},
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     doi = {10.1016/j.crma.2011.11.014},
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Martial Agueh. Finsler structure in the p-Wasserstein space and gradient flows. Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 35-40. doi : 10.1016/j.crma.2011.11.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.11.014/

Bibliographie
Cité par

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[3] J.-D. Benamou; Y. Brenier A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem, Numer. Math., Volume 84 (2000), pp. 375-393

[4] J.A. Carrillo; R.J. McCann; C. Villani Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., Volume 179 (2006), pp. 217-263

[5] W. Gangbo; R.J. McCann Optimal maps in the Mongeʼs mass transport problem, C. R. Acad. Sci. Paris, Ser. I, Volume 321 (1995), pp. 1653-1658

[6] M. Golomb; R.A. Tapia The metric gradient in normed linear spaces, Numer. Math., Volume 20 (1972), pp. 115-124

[7] R. Jordan; D. Kinderlehrer; F. Otto The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal., Volume 29 (1998), pp. 1-17

[8] R.J. McCann A convexity principle for interacting gases, Adv. Math., Volume 128 (1997), pp. 153-179

[9] F. Otto The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, Volume 26 (2001), pp. 101-174

[10] H. Rund The Differential Geometry of Finsler Spaces, Springer-Verlag, 1959

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