Ricci curvature of metric spaces

Ollivier, Yann

Comptes Rendus. Mathématique, Tome 345 (2007) no. 11, pp. 643-646.

Nous définissons la courbure de Ricci d'un espace métrique muni d'une mesure ou d'une marche aléatoire. Notre outil est un coefficient de contraction local de la marche aléatoire agissant sur l'espace des mesures de probabilités muni d'une distance de transport. Nous pouvons ainsi généraliser des résultats classiques en courbure de Ricci minorée, comme la borne sur le trou spectral (théorème de Lichnerowicz), la concentration gaussienne de la mesure (théorème de Lévy–Gromov), l'inégalité de Sobolev logarithmique (conséquence de la théorie de Bakry–Émery) ou le théorème de Bonnet–Myers. Notre définition est compatible avec la théorie de Bakry–Émery, est robuste, et très simple à mettre en œuvre concrètement, par exemple sur un graphe.

We define a notion of Ricci curvature in metric spaces equipped with a measure or a random walk. For this we use a local contraction coefficient of the random walk acting on the space of probability measures equipped with a transportation distance. This notions allows to generalize several classical theorems associated with positive Ricci curvature, such as a spectral gap bound (Lichnerowicz theorem), Gaussian concentration of measure (Lévy–Gromov theorem), logarithmic Sobolev inequalities (a result of Bakry–Émery theory) or the Bonnet–Myers theorem. The definition is compatible with Bakry–Émery theory, and is robust and very easy to implement in concrete examples such as graphs.

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