{
  "id": "1207.5003",
  "version": "v3",
  "published": "2012-07-20T17:03:29.000Z",
  "updated": "2015-03-16T20:22:51.000Z",
  "title": "Representation of Markov chains by random maps: existence and regularity conditions",
  "authors": [
    "Jürgen Jost",
    "Martin Kell",
    "Christian S. Rodrigues"
  ],
  "comment": "22 pages, several changes from the previous version including extended discussion of many details",
  "categories": [
    "math.DS",
    "math.DG",
    "math.PR"
  ],
  "abstract": "We systematically investigate the problem of representing Markov chains by families of random maps, and which regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. Optimal transport theory also tells us how convexity properties of the supports of the measures translate into regularity properties of the maps via Legendre transforms. Thus, from this scheme, we cannot only deduce the representation by measurable random maps, but we can also obtain conditions for the representation by continuous random maps. Finally, we present conditions for the representation of Markov chain by random diffeomorphisms.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-07-23T13:25:21.000Z",
      "abstract": "We systematically investigate the problem of representing Markov chains by families of random maps, and what regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. Optimal transport theory also tells us how convexity properties of the supports of the measures translate into regularity properties of the maps via Legendre transforms. Thus, from this scheme, we cannot not only deduce the representation by measurable random maps, but we can also obtain conditions for the representation by continuous random maps. Finally, we show how to construct random diffeomorphisms from a given Markov chain.",
      "comment": "17 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-03-16T20:22:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C40",
      "49K45",
      "49N60",
      "37H10",
      "37C05"
    ],
    "keywords": [
      "markov chain",
      "regularity conditions",
      "representation",
      "optimal transport theory",
      "construct random diffeomorphisms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.5003J"
    }
  }
}