Random walks and induced Dirichlet forms on self-similar sets

Abstract

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 X of K a natural augmented tree structure E exists; 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 M can 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 α is the Hausdorff dimension of K and β depends on λ. For suitable β, the kernel defines a regular non-local Dirichlet form on K. This extends the results of Kigami [27] concerning random walks on certain trees with Cantor-type sets as boundaries (see also [5]).