Critical exponents of induced Dirichlet forms on self-similar sets

Shi-Lei Kong; Ka-Sing Lau

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arXiv · 1612.01708

Critical exponents of induced Dirichlet forms on self-similar sets

Shi-Lei Kong·Ka-Sing Lau

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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 $\Lambda^{\alpha, \beta/2}_{2,2}$ 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 $\Lambda^{\alpha, \beta/2}_{2,2}$ 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.

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Shi-Lei Kong, Ka-Sing Lau. 2016-12-06. Critical exponents of induced Dirichlet forms on self-similar sets. https://arxiv.org/abs/1612.01708

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