Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds

Giuseppe Savaré

doi:10.1016/j.crma.2007.06.018

Functional Analysis

Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds
[Flots gradients dans des espaces métriques à courbure minorée]

Présenté par : Paul Malliavin

Giuseppe Savaré ¹

¹ Dipartimento di Matematica, Università di Pavia, Via Ferrata, 1, 27100 Pavia, Italy

Comptes Rendus. Mathématique, Volume 345 (2007) no. 3, pp. 151-154.

Résumé
Abstract

On présente dans cette Note quelques résultats nouveaux relatifs aux flots gradients associés aux fonctionnelles λ-convexes dans une large classe d'espaces métriques, comprenant les espaces d'Aleksandrov (à courbure minorée) et les espaces correspondants du type $L^{2}$ -Wasserstein. On considère aussi des applications aux flots gradients de l'entropie dans des espaces métriques mesurés à courbure de Ricci minorée et aux semigroupes de diffusion correspondants. Ces résultats ont été présentés au Congrés “Optimal Transport: theory and applications”, Pisa, Novembre 2006.

We present some new results concerning well-posedness of gradient flows generated by λ-convex functionals in a wide class of metric spaces, including Alexandrov spaces satisfying a lower curvature bound and the corresponding $L^{2}$ -Wasserstein spaces. Applications to the gradient flow of Entropy functionals in metric-measure spaces with Ricci curvature bounded from below and to the corresponding diffusion semigroup are also considered. These results have been announced during the workshop on “Optimal Transport: theory and applications” held in Pisa, November 2006.

Reçu le : 2006-12-12
Accepté le : 2007-06-14
Publié le : 2007-07-27

DOI : 10.1016/j.crma.2007.06.018

Affiliations des auteurs :

Giuseppe Savaré ¹

¹ Dipartimento di Matematica, Università di Pavia, Via Ferrata, 1, 27100 Pavia, Italy

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Giuseppe Savaré. Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds. Comptes Rendus. Mathématique, Volume 345 (2007) no. 3, pp. 151-154. doi : 10.1016/j.crma.2007.06.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.06.018/

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