Comptes Rendus
no. 3
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 Savaré1
1 Dipartimento di Matematica, Università di Pavia, Via Ferrata, 1, 27100 Pavia, Italy
Comptes Rendus. Mathématique, Volume 345 (2007) no. 3, pp. 151-154.

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 L2-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 L2-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 :
Accepté le :
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DOI : 10.1016/j.crma.2007.06.018
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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[3] H. Brézis Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, No. 5, North-Holland Publishing Co., Amsterdam, 1973

[4] D. Burago; Y. Burago; S. Ivanov A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001

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

[6] J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. Math., in press

[7] U.F. Mayer Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom., Volume 6 (1998), pp. 199-253

[8] S.-I. Ohta, Gradient flows on Wasserstein spaces over compact Alexandrov spaces, preprint, 2007

[9] G. Perelman, A. Petrunin, Quasigeodesics and gradient curves in Alexandrov spaces, unpublished manuscript

[10] K.-T. Sturm On the geometry of metric measure spaces. I, Acta Math., Volume 196 (2006), pp. 65-131

[11] C. Villani Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003

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