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Random Walks on Sierpiński Graphs: Hyperbolicity and Stochastic Homogenization

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Fractals in Graz 2001

Part of the book series: Trends in Mathematics ((TM))

Abstract

We introduce two new techniques to the analysis on fractals. One is based on the presentation of the fractal as the boundary of a countable Gromov hyperbolic graph, whereas the other one consists in taking all possible “backward” extensions of the above hyperbolic graph and considering them as the classes of a discrete equivalence relation on an appropriate compact space. Illustrating these techniques on the example of the Sierpiński gasket (the associated hyperbolic graph is called the Sierpiński graph), we show that the Sierpiński gasket can be identified with the Martin and the Poisson boundaries for fairly general classes of Markov chains on the Sierpiński graph.

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Author information

Authors and Affiliations

  1. CNRS UMR 6625 (IRMAR), Université Rennes-1 Campus Beaulieu, 35042, Rennes, France

    Vadim A. Kaimanovich

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  1. Vadim A. Kaimanovich
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Editors and Affiliations

  1. Institut für Mathematik A, Technische Universität Graz, Steyrergasse 30, 8010, Graz, Austria

    Peter Grabner

  2. Institut für Mathematik C, Technische Universität Graz, Steyrergasse 30, 8010, Graz, Austria

    Wolfgang Woess

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Kaimanovich, V.A. (2003). Random Walks on Sierpiński Graphs: Hyperbolicity and Stochastic Homogenization. In: Grabner, P., Woess, W. (eds) Fractals in Graz 2001. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8014-5_5

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  • DOI: https://doi.org/10.1007/978-3-0348-8014-5_5

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9403-6

  • Online ISBN: 978-3-0348-8014-5

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