{
  "id": "math/0703451",
  "version": "v1",
  "published": "2007-03-15T12:46:03.000Z",
  "updated": "2007-03-15T12:46:03.000Z",
  "title": "Trends to Equilibrium in Total Variation Distance",
  "authors": [
    "Patrick Cattiaux",
    "Arnaud Guillin"
  ],
  "comment": "36 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a given integrability condition. To this end, we give a general upper bound \"\\`{a} la Pinsker\" enabling us to study our problem firstly via usual functional inequalities (Poincar\\'{e} inequality, weak Poincar\\'{e},...) and truncation procedure, and secondly through the introduction of new functional inequalities $\\Ipsi$. These $\\Ipsi$-inequalities are characterized through measure-capacity conditions and $F$-Sobolev inequalities. A direct study of the decay of Hellinger distance is also proposed. Finally we show how a dynamic approach based on reversing the role of the semi-group and the invariant measure can lead to interesting bounds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-03-15T12:46:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26D10",
      "60E15"
    ],
    "keywords": [
      "total variation distance",
      "inequality",
      "equilibrium",
      "ergodic diffusion processes",
      "general upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3451C"
    }
  }
}