Random walks and induced Dirichlet forms on compact spaces of homogeneous type

TL;DR

This paper extends the analysis of random walks and Dirichlet forms from self-similar sets to more general compact spaces of homogeneous type, establishing a connection between random walks and energy forms on these spaces.

Contribution

It introduces a class of transient reversible random walks on hyperbolic augmented trees associated with homogeneous type spaces and characterizes the induced energy forms.

Findings

The induced energy form is comparable to a double integral involving a kernel with a specific decay.

For $ ext{alpha}$-sets in $ ext{R}^d$, the energy kernel behaves like $|\xi-\eta|^{- ext{alpha}-eta}$.

Conditions are discussed under which the energy form becomes a non-local regular Dirichlet form.

Abstract

We extend our study of random walks and induced Dirichlet forms on self-similar sets [arXiv:1604.05440, 1612.01708] to compact spaces of homogeneous type $(K, \rho ,\mu)$. A successive partition on $K$ brings a natural augmented tree structure $(X, E)$ that is Gromov hyperbolic, and the hyperbolic boundary is H\"older equivalent to $K$. We then introduce a class of transient reversible random walks on $(X, E)$ with return ratio $\lambda$. Using Silverstein's theory of Markov chains, we prove that the random walk induces an energy form on $K$ with $$ {\mathcal E}_K [u] \asymp \iint_{K\times K \setminus \Delta} \frac{|u(\xi) - u(\eta)|^2}{V(\xi, \eta)\rho (\xi, \eta)^\beta} d\mu(\xi) d\mu(\eta), $$ where $V(\xi, \eta)$ is the $\mu$-volume of the ball centered at $\xi$ with radius $\rho (\xi, \eta)$, $\Delta$ is the diagonal, and $\beta$ depends on $\lambda$. In particular, for an $\alpha$-set in ${\mathbb R}^d$, the kernel of the energy form is of order $\frac{1}{|\xi-\eta|^{\alpha +\beta}}$. We also discuss conditions for this energy form to be a non-local regular Dirichlet form.