{
  "id": "1306.6180",
  "version": "v2",
  "published": "2013-06-26T09:31:46.000Z",
  "updated": "2015-12-09T14:26:07.000Z",
  "title": "Discrete random walks on the group Sol",
  "authors": [
    "Jérémie Brieussel",
    "Ryokichi Tanaka"
  ],
  "journal": "Israel Journal of Mathematics, September 2015, Volume 208, Issue 1, pp 291-321",
  "categories": [
    "math.PR",
    "math.GR"
  ],
  "abstract": "The harmonic measure $\\nu$ on the boundary of the group $Sol$ associated to a discrete random walk of law $\\mu$ was described by Kaimanovich. We investigate when it is absolutely continuous or singular with respect to Lebesgue measure. By ratio entropy over speed, we show that any countable non-abelian subgroup admits a finite first moment non-degenerate $\\mu$ with singular harmonic measure $\\nu$. On the other hand, we prove that some random walks with finitely supported step distribution admit a regular harmonic measure. Finally, we construct some exceptional random walks with arbitrarily small speed but singular harmonic measures. The two later results are obtained by comparison with Bernoulli convolutions, using results of Erd\\\"os and Solomyak.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-26T09:31:46.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-12-09T14:26:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "22E25",
      "43A80",
      "22E40"
    ],
    "keywords": [
      "discrete random walk",
      "group sol",
      "supported step distribution admit",
      "singular harmonic measure",
      "finite first moment non-degenerate"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.6180B"
    }
  }
}