Electronic Journal of Probability

Random Walks on Groups and Monoids with a Markovian Harmonic Measure

Mairesse Jean

Full-text: Open access

Abstract

We consider a transient nearest neighbor random walk on a group $G$ with finite set of generators $S$. The pair $(G,S)$ is assumed to admit a natural notion of normal form words where only the last letter is modified by multiplication by a generator. The basic examples are the free products of a finitely generated free group and a finite family of finite groups, with natural generators. We prove that the harmonic measure is Markovian of a particular type. The transition matrix is entirely determined by the initial distribution which is itself the unique solution of a finite set of polynomial equations of degree two. This enables to efficiently compute the drift, the entropy, the probability of ever hitting an element, and the minimal positive harmonic functions of the walk. The results extend to monoids.

Article information

Source
Electron. J. Probab., Volume 10 (2005), paper no. 43, 1417-1441.

Dates
Accepted: 16 December 2005
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464816844

Digital Object Identifier
doi:10.1214/EJP.v10-293

Mathematical Reviews number (MathSciNet)
MR2191634

Zentralblatt MATH identifier
1109.60037

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 31C05: Harmonic, subharmonic, superharmonic functions 60J22: Computational methods in Markov chains [See also 65C40] 65C40: Computational Markov chains 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

Keywords
Finitely generated group or monoid free product random walk harmonic measure

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Jean, Mairesse. Random Walks on Groups and Monoids with a Markovian Harmonic Measure. Electron. J. Probab. 10 (2005), paper no. 43, 1417--1441. doi:10.1214/EJP.v10-293. https://projecteuclid.org/euclid.ejp/1464816844


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