{
  "id": "2203.02534",
  "version": "v1",
  "published": "2022-03-04T19:21:58.000Z",
  "updated": "2022-03-04T19:21:58.000Z",
  "title": "Discrete self-similar and ergodic Markov chains",
  "authors": [
    "Laurent Miclo",
    "Pierre Patie",
    "Rohan Sarkar"
  ],
  "comment": "50 pages",
  "journal": "Annals of Probability, 2022",
  "categories": [
    "math.PR",
    "math.FA",
    "math.SP"
  ],
  "abstract": "The first aim of this paper is to introduce a class of Markov chains on $\\mathbb{Z}_+$ which are discrete self-similar in the sense that their semigroups satisfy an invariance property expressed in terms of a discrete random dilation operator. After showing that this latter property requires the chains to be upward skip-free, we first establish a gateway relation, a concept introduced in [26], between the semigroup of such chains and the one of spectrally negative self-similar Markov processes on $\\mathbb{R}_+$. As a by-product, we prove that each of these Markov chains, after an appropriate scaling, converge in the Skorohod metric, to the associated self-similar Markov process. By a linear perturbation of the generator of these Markov chains, we obtain a class of ergodic Markov chains, which are non-reversible. By means of intertwining and interweaving relations, where the latter was recently introduced in [27], we derive several deep analytical properties of such ergodic chains including the description of the spectrum, the spectral expansion of their semigroups, the study of their convergence to equilibrium in the $\\Phi$-entropy sense as well as their hypercontractivity property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-04T19:21:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A60",
      "47G20",
      "33C45",
      "47D07",
      "37A30",
      "60J27",
      "60J60"
    ],
    "keywords": [
      "ergodic markov chains",
      "discrete self-similar",
      "discrete random dilation operator",
      "spectrally negative self-similar markov processes",
      "associated self-similar markov process"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}