Random walks and induced Dirichlet forms on self-similar sets
Publication in refereed journal


摘要Let 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]).
著者Shi-Lei Kong, Ka-Sing Lau, Ting-Kam Leonard Wong
期刊名稱Advances in Mathematics
出版社Elsevier Inc.
頁次1099 - 1134

上次更新時間 2021-22-02 於 00:48