Ricci curvature of Markov chains on metric spaces

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dc.creatorOllivier, Yann
dc.date2007-01-30
dc.date2007-07-30
dc.date.accessioned2026-07-07T08:20:49Z
dc.date.available2026-07-07T08:20:49Z
dc.descriptionWe define the Ricci curvature of Markov chains on metric spaces as a local contraction coefficient of the random walk acting on the space of probability measures equipped with a Wasserstein transportation distance. For Brownian motion on a Riemannian manifold this gives back the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein--Uhlenbeck process. Moreover this generalization is consistent with the Bakry--Émery Ricci curvature for Brownian motion with a drift on a Riemannian manifold. Positive Ricci curvature is easily shown to imply a spectral gap, a Lévy--Gromov-like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. These bounds are sharp in several interesting examples.
dc.identifierhttps://arxiv.org/abs/math/0701886
dc.identifierhttp://arxiv.org/abs/math/0701886
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/135179
dc.subjectProbability
dc.subjectMetric Geometry
dc.subject51F99, 53B21, 60B99
dc.titleRicci curvature of Markov chains on metric spaces
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