{
  "id": "math/0701886",
  "version": "v4",
  "published": "2007-01-30T16:41:12.000Z",
  "updated": "2007-07-30T17:57:20.000Z",
  "title": "Ricci curvature of Markov chains on metric spaces",
  "authors": [
    "Yann Ollivier"
  ],
  "categories": [
    "math.PR",
    "math.MG"
  ],
  "abstract": "We 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--\\'Emery Ricci curvature for Brownian motion with a drift on a Riemannian manifold. Positive Ricci curvature is easily shown to imply a spectral gap, a L\\'evy--Gromov-like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. These bounds are sharp in several interesting examples.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2007-07-30T17:57:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51F99",
      "53B21",
      "60B99"
    ],
    "keywords": [
      "markov chains",
      "metric spaces",
      "brownian motion",
      "riemannian manifold",
      "wasserstein transportation distance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1886O"
    }
  }
}