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Denker–Sato type Markov chains on self-similar sets

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Abstract

In Denker and Sato (Potential Anal 14: 211–232, 2001; Publ RIMS 35: 769–794, 1999; Math Nachr 241: 32–55, 2002) studied certain Markov chain on the symbolic spaces of the Sierpinski gasket (SG). They showed that the Martin boundary is homeomorphic to SG, and used the potential theory on the Martin boundary to induce a harmonic structure on SG. In this paper, we consider a more general Denker–Sato type Markov chain associated with self-similar sets \(K\) with the open set condition. The chain is defined on the augmented tree of the symbolic space. Such tree was introduced by Kaimanovich, it is hyperbolic in the sense of Gromov (Kaimanovich in Random walks on Sierpiński graphs: hyperbolicity and stochastic homogenization, Birha̋user, Basel, 2003; Lau and Wang in Indiana Univ Math J 58:1777–1795, 2009). We show that the Martin boundary, the hyperbolic boundary and the self-similar set \(K\) are homeomorphic. The hitting distribution of the chain is also obtained.

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Acknowledgments

The authors are grateful for the helpful discussions with Professors D. J. Feng, H. Rao, H. J. Ruan and Ms. H. Koivusalo. Part of this work was carried out while the second author was visiting the Department of Mathematics of the Chinese University of Hong Kong, he thanks the Department for the support.

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Authors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

    Ka-Sing Lau

  2. School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou, 510275, People’s Republic of China

    Xiang-Yang Wang

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  1. Ka-Sing Lau
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  2. Xiang-Yang Wang
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Correspondence to Xiang-Yang Wang.

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The research is supported in part by the HKRGC Grant, and the NNSF of China (No. 11171100, 11371382).

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Lau, KS., Wang, XY. Denker–Sato type Markov chains on self-similar sets. Math. Z. 280, 401–420 (2015). https://doi.org/10.1007/s00209-015-1430-y

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