{
  "id": "1012.5347",
  "version": "v2",
  "published": "2010-12-24T06:54:31.000Z",
  "updated": "2011-09-02T02:53:08.000Z",
  "title": "Induced measures of simple random walks on Sierpinski graphs",
  "authors": [
    "Ting Kam Leonard Wong"
  ],
  "comment": "19 pages, 10 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "In \\cite{[K]}, Kaimanovich defined an augmented rooted tree $(X, E)$ corresponding to the Sierpinski gasket $K$, and showed that the Martin boundary of the simple random walk $\\{Z_n\\}$ on it is homeomorphic to $K$. It is of interest to determine the hitting distributions $v_{{\\bf x}}(\\cdot) = {\\mathbb{P}}_{{\\bf x}}\\{\\lim_{n \\rightarrow \\infty} Z_n \\in \\cdot\\}$ induced on $K$. Using a reflection principle based on the symmetries of $K$, we show that if the walk starts at the root of $(X, E)$, the hitting distribution is exactly the normalized Hausdorff measure $\\mu$ on $K$. In particular, each $v_{{\\bf x}}$, ${\\bf x} \\in X$, is absolutely continuous with respect to $\\mu$. This answers a question of Kaimanovich [K, Problem 4.14]. The argument can be generalized to other symmetric self-similar sets.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-09-02T02:53:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple random walk",
      "sierpinski graphs",
      "induced measures",
      "hitting distribution",
      "symmetric self-similar sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.5347W"
    }
  }
}