Random walks and induced Dirichlet forms on self-similar sets

TL;DR

This paper studies reversible random walks on self-similar sets, establishing their boundary identifications, kernel estimates, and associated non-local Dirichlet forms, extending Kigami's work on tree-based fractals.

Contribution

It introduces a framework connecting random walks on augmented trees to Dirichlet forms on self-similar sets, with explicit kernel estimates and boundary identifications.

Findings

Martin boundary identified with hyperbolic boundary and fractal set

Explicit estimates of Martin and Na"{i}m kernels in terms of Gromov product

Na"{i}m kernel defines a non-local Dirichlet form on the fractal

Abstract

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 ${\mathfrak E}$ exists; it is hyperbolic, and the hyperbolic boundary $\partial_HX$ 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 $(X, {\mathfrak E})$. 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\"{i}m kernel in terms of the Gromov product. Moreover, the Na\"{i}m 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.