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

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 Hölder 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.

21 pages, no figures