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Transactions of the American Mathematical Society

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Local and non-local Dirichlet forms on the Sierpiński carpet


Authors: Alexander Grigor’yan and Meng Yang
Journal: Trans. Amer. Math. Soc. 372 (2019), 3985-4030
MSC (2010): Primary 28A80
DOI: https://doi.org/10.1090/tran/7753
Published electronically: May 9, 2019
MathSciNet review: 4009425
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Abstract: We give a purely analytic construction of a self-similar local regular Dirichlet form on the Sierpiński carpet using approximation of stable-like non-local closed forms which gives an answer to an open problem in analysis on fractals.


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Additional Information

Alexander Grigor’yan
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
Email: grigor@math.uni-bielefeld.de

Meng Yang
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Address at time of publication: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
Email: ymeng@math.uni-bielefeld.de, meng-yang13@mails.tsinghua.edu.cn

DOI: https://doi.org/10.1090/tran/7753
Keywords: Sierpi\'nski carpet, non-local quadratic form, walk dimension, $\Gamma$-convergence, Brownian motion, effective resistance, heat kernel
Received by editor(s): August 21, 2017
Received by editor(s) in revised form: July 12, 2018
Published electronically: May 9, 2019
Additional Notes: The authors were supported by SFB701 and SFB1283 of the German Research Council (DFG)
The second author is the corresponding author
Article copyright: © Copyright 2019 American Mathematical Society