Random walks and induced Dirichlet forms on self-similar sets
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AbstractLet K be a self-similar set satisfying the open set condition. Following Kaimanovich’s elegant idea [25], it has been proved that on the symbolic space Xof Ka natural augmented tree structure Eexists; it is hyperbolic, and the hyperbolic boundary ∂HX with the Gromov metric is Hölder equivalent to K. In this paper we consider certain reversible random walks with return ratio 0 <λ <1 on (X, E). We show that the Martin boundary Mcan be identified with ∂HX and K. With this setup and a device of Silverstein[41], we obtain precise estimates of the Martin kernel and the Naïm kernel in terms of the Gromov product. Moreover, the Naïm kernel turns out to be a jump kernel satisfying the estimate Θ(ξ, η) |ξ−η|−(α+β), where αis the Hausdorff dimension of Kand βdepends on λ. For suitable β, the kernel defines a regular non-local Dirichlet form onK. This extends the results of Kigami [27] concerning random walks on certain trees with Cantor-type sets as boundaries (see also [5]).
All Author(s) ListShi-Lei Kong, Ka-Sing Lau, Ting-Kam Leonard Wong
Journal nameAdvances in Mathematics
Year2017
Month11
Day7
Volume Number320
PublisherElsevier Inc.
Pages1099 - 1134
ISSN0001-8708
eISSN1090-2082
LanguagesEnglish-United States

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