Random walks and induced Dirichlet forms on self-similar sets

Shi-Lei Kong The Chinese University of Hong Kong Ka-Sing Lau The Chinese University of Hong Kong Ting-Kam Leonard Wong University of Southern California

Dynamical Systems Functional Analysis Metric Geometry Probability mathscidoc:1701.28005

Advances in Mathematics, 320, 1099--1134, 2017.9
Let $K$ be a self-similar set satisfying the open set condition. Following Kaimanovich’s elegant idea, it has been proved that on the symbolic space $X$ of $K$ a natural augmented tree structure $E$ exists; it is hyperbolic, and the hyperbolic boundary $\partial_H X$ with the Gromov metric is H\"older equivalent to $K$. In this paper we consider certain reversible random walks with return ratio $0 < \lambda < 1$ on $(X,E)$. We show that the Martin boundary ${\mathcal M}$ can be identified with $\partial_H X$ and $K$. With this setup and a device of Silverstein, we obtain precise estimates of the Martin kernel and the Na\"im kernel in terms of the Gromov product. Moreover, the Naïm kernel turns out to be a jump kernel satisfying the estimate $\Theta (\xi,\eta) \asymp |\xi-\eta|^{-(\alpha+\beta)}$, where $\alpha$ is the Hausdorff dimension of $K$ and $\beta$ depends on $\lambda$. For suitable $\beta$, the kernel defines a regular non-local Dirichlet form on $K$. This extends the results of Kigami concerning random walks on certain trees with Cantor-type sets as boundaries.
Dirichlet form, hyperbolic boundary, Martin boundary, Na\"im kernel, self-similar set, reversible random walk.
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@inproceedings{shi-lei2017random,
  title={Random walks and induced Dirichlet forms on self-similar sets},
  author={Shi-Lei Kong, Ka-Sing Lau, and Ting-Kam Leonard Wong},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170113110154620765099},
  booktitle={Advances in Mathematics},
  volume={320},
  pages={1099--1134},
  year={2017},
}
Shi-Lei Kong, Ka-Sing Lau, and Ting-Kam Leonard Wong. Random walks and induced Dirichlet forms on self-similar sets. 2017. Vol. 320. In Advances in Mathematics. pp.1099--1134. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170113110154620765099.
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