The Sierpi\'nski gasket as the Martin boundary of a non-isotropic Markov chain

TL;DR

This paper extends the understanding of Martin boundaries by analyzing a class of non-isotropic Markov chains, showing they have a Sierpiński gasket as their Martin boundary and describing their harmonic functions.

Contribution

It generalizes previous results from isotropic to non-isotropic Markov chains, identifying the Martin boundary as a Sierpiński gasket and characterizing harmonic functions.

Findings

Martin boundary is homeomorphic to the Sierpiński gasket

Minimal Martin boundary is a proper subset of the Martin boundary

Provides a description of harmonic functions for the chain

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\'nski 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.