Random walks and induced Dirichlet forms on self-similar sets

Let $K$ be a self-similar set with the open set condition. It is known that there is a naturally defined augmented tree structure $E$ on the symbolic space $X$ of $K$ that is hyperbolic, and the hyperbolic boundary ${\partial}_H X$ with the Gromov metric is Holder equivalent to $K$. In the paper we consider certain reversible random walks with return ratio $0 < {\lambda} < 1$ on $(X,E)$. 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 are able to obtain precise estimates of the Martin kernel and the Naim kernel in terms of the Gromov product, and the Naim kernel turns out to be a jump kernel ${\Theta}({\xi}, {\eta}) {\asymp} |{\xi} - {\eta}|^{-({\alpha}+{\beta})}$ where ${\alpha}$ is the Hausdorff dimension of $K$ and ${\beta}$ depends on ${\lambda}$. For suitable ${\beta}$, the kernel defines a non-local Dirichlet form on $K$. This extends a consideration of Kigami, where he investigated random walks on the binary trees with Cantor type sets as boundaries.