Electronic Journal of Probability

Random Walks on Groups and Monoids with a Markovian Harmonic Measure

Mairesse Jean

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Finitely generated group or monoid free product random walk harmonic measure

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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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