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Brownian motion on the golden ratio Sierpinski gasket

Part of: Potential theory on metric spaces Classical measure theory

Published online by Cambridge University Press:  03 April 2023

Shiping Cao  and
Hua Qiu
Shiping Cao
Affiliation:
Department of Mathematics, Cornell University, Ithaca 14853, USA sc2873@cornell.edu
Hua Qiu
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, China huaqiu@nju.edu.cn
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Abstract

We construct a strongly local regular Dirichlet form on the golden ratio Sierpinski gasket, which is a self-similar set without a finitely ramified cell structure, via a study on the trace of an electrical network on an infinite graph. The Dirichlet form is the unique one that is self-similar in the sense of an infinite iterated function system, and is decimation invariant with respect to a graph-directed construction. The proof is based on a fixed point problem of a renormalization map, inspired by Sabot's celebrated work for finitely ramified fractals. Lastly, the Hunt process associated with the Dirichlet form satisfies a two-sided sub-Gaussian heat kernel estimate.

Keywords

golden ratio Sierpinski gasketinfinite graphDirichlet formsheat kernel estimates

MSC classification

Primary: 28A80: Fractals
Secondary: 31E05: Potential theory on metric spaces
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 28
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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