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, 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 ${\partial }_{H}X$ with the Gromov metric is Hölder 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 Naïm kernel in terms of the Gromov product. Moreover, the Naïm 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.

33 pages with 2 figures

References in corpus (2)

• Critical exponents of induced Dirichlet forms on self-similar sets
• Determination of the Walk Dimension of the Sierpiński Gasket Without Using Diffusion

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