Critical exponents of induced Dirichlet forms on self-similar sets

Shi-Lei Kong The Chinese University of Hong Kong Ka-Sing Lau The Chinese University of Hong Kong

Functional Analysis mathscidoc:1701.12031

2016.10
In a previous paper, 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 $K$. 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 consider 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 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.
Besov space, Dirichlet form, hyperbolic graph, minimal energy, Martin boundary, resistance, self-similar set, random walk.
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@inproceedings{shi-lei2016critical,
  title={Critical exponents of induced Dirichlet forms on self-similar sets},
  author={Shi-Lei Kong, and Ka-Sing Lau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170113110912600977100},
  year={2016},
}
Shi-Lei Kong, and Ka-Sing Lau. Critical exponents of induced Dirichlet forms on self-similar sets. 2016. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170113110912600977100.
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