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 ∂ H X 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 ∂ H X 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]).

Dirichlet forms and critical exponents on fractals

“…set), and the induced Dirichlet form on the boundary has the expression in (7.4). In such setting the critical exponents of the B σ 2,2 in connection with the random walk has also been studied in [22].…”

“…It is well-known that if K is a domain in R d , then σ * = 1; if K is the d-dimensional Sierpinski gasket, then σ * = log(d + 3)/(2 log 2) [17]. There are extensions on the nested fractals [27], and approximate value of the Sierpinski carpet [2]; also for Cantor-type sets, σ * = ∞ [21]. More generally, the notion of Besov space and critical exponents have been extended to metric measure spaces (K, d, µ), where (K, d) is a locally compact, separable metric space, and µ is α-regular as before.…”

On a recursive construction of Dirichlet form on the Sierpiński gasket

“…The original usage of the above mentioned p.c.f sets was to study the critical exponents σ * of the Besov spaces B σ 2,∞ ⊂ L 2 (K, µ), σ > 0 (where K ⊂ R d is closed, and µ is an α-Ahlfors regular measure on K) in connection with the domain of the Dirichlet forms and the walk dimension ( [8,20,21] |u(x) − u(y)| 2 dµ(y) dµ(x).…”

Self‐similar sets, simple augmented trees and their Lipschitz equivalence

“…Recently, they [19] completed the previous studies by removing some superfluous conditions, and obtained that for any IFS, the augmented tree is always a hyperbolic graph. Moreover, the hyperbolic boundary is Hölder equivalent to the K. The setup of augmented tree connects fractal geometry, graph theory and Markov chains, it has been frequently used to study the random walks on self-similar sets and Dirichlet forms [11,[13][14][15]18]. On the other hand, the author and his collaborators made a first attempt to apply the augmented tree to the study on Lipschitz equivalence of self-similar sets.…”