The Sierpiński gasket as the Martin boundary of a non-isotropic Markov chain

Marc Kesseböhmer; Tony Samuel; Karenina Sender

doi:10.4171/JFG/86

The Sierpiński gasket as the Martin boundary of a non-isotropic Markov chain

Marc Kesseböhmer, Tony Samuel, Karenina Sender

Mathematics

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Abstract

In 2012 Lau and Ngai, motivated by the work of Denker and Sato, gave an example of an isotropic Markov chain on the set of finite words over a three letter alphabet, whose Martin boundary is homeomorphic to the Sierpiński gasket. Here, we extend the results of Lau and Ngai to a class of non-isotropic Markov chains. We determine the Martin boundary and show that the minimal Martin boundary is a proper subset of the Martin boundary. In addition, we give a description of the set of harmonic functions.

Original language	English
Pages (from-to)	113-136
Number of pages	21
Journal	Journal of Fractal Geometry
Volume	7
Issue number	2
Early online date	19 May 2020
DOIs	https://doi.org/10.4171/JFG/86
Publication status	E-pub ahead of print - 19 May 2020

Keywords

Green function
Harmonic function
Markov chain
Martin boundary
Sierpinski gasket

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Kessebohmer_et_al_The_Sierpinski_gasket_Journal_of_Fractal_Geometry_2018Accepted author manuscript, 228 KBLicence: None: All rights reserved

https://www.ems-ph.org/journals/show_abstract.php?issn=2308-1309&vol=7&iss=2&rank=1Licence: None: All rights reserved

Cite this

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abstract = "In 2012 Lau and Ngai, motivated by the work of Denker and Sato, gave an example of an isotropic Markov chain on the set of finite words over a three letter alphabet, whose Martin boundary is homeomorphic to the Sierpi{\'n}ski gasket. Here, we extend the results of Lau and Ngai to a class of non-isotropic Markov chains. We determine the Martin boundary and show that the minimal Martin boundary is a proper subset of the Martin boundary. In addition, we give a description of the set of harmonic functions. ",

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AU - Sender, Karenina

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N2 - In 2012 Lau and Ngai, motivated by the work of Denker and Sato, gave an example of an isotropic Markov chain on the set of finite words over a three letter alphabet, whose Martin boundary is homeomorphic to the Sierpiński gasket. Here, we extend the results of Lau and Ngai to a class of non-isotropic Markov chains. We determine the Martin boundary and show that the minimal Martin boundary is a proper subset of the Martin boundary. In addition, we give a description of the set of harmonic functions.

AB - In 2012 Lau and Ngai, motivated by the work of Denker and Sato, gave an example of an isotropic Markov chain on the set of finite words over a three letter alphabet, whose Martin boundary is homeomorphic to the Sierpiński gasket. Here, we extend the results of Lau and Ngai to a class of non-isotropic Markov chains. We determine the Martin boundary and show that the minimal Martin boundary is a proper subset of the Martin boundary. In addition, we give a description of the set of harmonic functions.

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KW - Harmonic function

KW - Markov chain

KW - Martin boundary

KW - Sierpinski gasket

UR - https://arxiv.org/abs/1710.04414

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JO - Journal of Fractal Geometry

JF - Journal of Fractal Geometry

IS - 2

ER -

The Sierpiński gasket as the Martin boundary of a non-isotropic Markov chain

Abstract

Keywords

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