Comptes Rendus
no. 3
Functional Analysis
Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds
Paul Malliavin
Giuseppe Savaré1
1Dipartimento di Matematica, Università di Pavia, Via Ferrata, 1, 27100 Pavia, Italy
Comptes Rendus. Mathématique, Volume 345 (2007) no. 3, pp. 151-154

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.06.018

Giuseppe Savaré  1

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