Critical exponents of induced Dirichlet forms on self-similar sets

Abstract

In a previous paper [arXiv:1604.05440], we studied certain random walks on the hyperbolic graphs $X$ associated with the self-similar sets $K$, and showed that the discrete energy ${\mathcal{E}}_X$ on $X$ has an induced energy form ${\mathcal{E}}_K$ on $K$ that is a Gagliardo-type integral. The domain of ${\mathcal{E}}_K$ is a Besov space ${\Lambda }_{2,2}^{\alpha ,\beta /2}$ where $\alpha$ is the Hausdorff dimension of $K$ and $\beta$ is a parameter determined by the "return ratio" of the random walk. In this paper, we study the functional relationship of ${\mathcal{E}}_X$ and ${\mathcal{E}}_K$. In particular, we investigate the critical exponents of the $\beta$ in the domain ${\Lambda }_{2,2}^{\alpha ,\beta /2}$ in order for ${\mathcal{E}}_K$ to be a regular Dirichlet form. We provide some criteria to determine the critical exponents through the effective resistance of the random walk on $X$, and make use of certain electrical network techniques to calculate the exponents for some concrete examples.

40 pages, 10 figures

References in corpus (3)

• Random walks and induced Dirichlet forms on self-similar sets
• Self-Similar Jordan Arcs Which Do Not Satisfy OSC
• Dirichlet forms and critical exponents on fractals

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