{
  "id": "1102.5238",
  "version": "v2",
  "published": "2011-02-25T13:19:45.000Z",
  "updated": "2011-06-16T09:07:21.000Z",
  "title": "Gradient flows of the entropy for finite Markov chains",
  "authors": [
    "Jan Maas"
  ],
  "comment": "An error in Example 2.6 has been corrected and several changes have been made accordingly. To appear in J. Funct. Anal",
  "categories": [
    "math.PR",
    "math.CA",
    "math.DG",
    "math.MG"
  ],
  "abstract": "Let K be an irreducible and reversible Markov kernel on a finite set X. We construct a metric W on the set of probability measures on X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy. This result is a discrete counterpart of the Wasserstein gradient flow interpretation of the heat flow in R^n by Jordan, Kinderlehrer, and Otto (1998). The metric W is similar to, but different from, the L^2-Wasserstein metric, and is defined via a discrete variant of the Benamou-Brenier formula.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-06-16T09:07:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J27",
      "28A33",
      "49Q20",
      "60B10"
    ],
    "keywords": [
      "finite markov chains",
      "continuous time markov chain evolves",
      "wasserstein gradient flow interpretation",
      "benamou-brenier formula",
      "probability measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.5238M"
    }
  }
}