Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

General random walk in a random environment defined on Galton–Watson trees

A. D. Barbour and Andrea Collevecchio

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Abstract

We consider a particle performing a random walk on a Galton–Watson tree, when the probabilities of jumping from a vertex to any one of its neighbours are determined by a random process. We introduce a method for deriving conditions under which the walk is either transient or recurrent. We first suppose that the weights are i.i.d., and re-prove a result of Lyons and Pemantle (Ann. Probab. 20 (1992) 125–136). We then assume a Markovian environment along each line of descent, and finally consider a random walk in a Markovian environment that itself changes the environment. Our approach involves studying the typical behaviour of the walk on fixed lines of descent, which we then show determines the behaviour of the process on the whole tree.

Résumé

Nous considérons le mouvement d’une particule sur un arbre de Galton–Watson, lorsque les probabilités de saut d’un nœud à l’un de ses voisins sont déterminées par un processus aléatoire. Conditionnellement à l’arbre, des poids positifs sont affectés aux arcs de telle sorte que, vu le long d’une ligne de descente, ils évoluent comme un processus aléatoire. Afin de présenter notre méthode pour prouver la récurrence ou la transience du processus, nous supposons d’abord que les poids sont i.i.d., redémontrant ainsi un résultat de Lyons et Pemantle. Nous étendons ensuite l’argument pour permettre un environnement Markovien, et enfin à une marche aléatoire sur un environnement Markovien qui modifie l’environnement. Notre approche consiste à étudier le comportement typique des processus sur les lignes de descente fixes, dont nous montrons ensuite qu’il détermine le comportement du processus sur l’ensemble de l’arbre.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1657-1674.

Dates
Received: 14 October 2015
Revised: 29 April 2016
Accepted: 16 May 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773722

Digital Object Identifier
doi:10.1214/16-AIHP766

Mathematical Reviews number (MathSciNet)
MR3729631

Zentralblatt MATH identifier
1382.60124

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Random walk in random environment Galton–Watson Reinforcement

Citation

Barbour, A. D.; Collevecchio, Andrea. General random walk in a random environment defined on Galton–Watson trees. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1657--1674. doi:10.1214/16-AIHP766. https://projecteuclid.org/euclid.aihp/1511773722


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References

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