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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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


Export citation

References

  • [1] A. Collevecchio. On the transience of processes defined on Galton–Watson trees. Ann. Probab. 34 (2006) 870–878.
  • [2] J. J. Dai. A once edge-reinforced random walk on a Galton–Watson tree is transient. Statist. Probab. Lett. 73 (2005) 115–124.
  • [3] A. Dembo and T. Zajic. Large deviations: From empirical mean and measure to partial sums process. Stochastic Process. Appl. 57 (1995) 191–224.
    Mathematical Reviews (MathSciNet): MR1334969
    Digital Object Identifier: doi:10.1016/0304-4149(94)00048-X
  • [4] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, Berlin, 1998.
  • [5] J.-D. Deuschel and D. W. Stroock. Large Deviations. Academic Press, Boston, MA, 1989.
  • [6] R. Durrett, H. Kesten and V. Limic. Once edge-reinforced random walk on a tree. Probab. Theory Related Fields 122 (2002) 567–592.
    Mathematical Reviews (MathSciNet): MR1902191
    Digital Object Identifier: doi:10.1007/s004400100179
  • [7] G. Faraud. A central limit theorem for random walk in a random environment on marked Galton–Watson trees. Electron. J. Probab. 16 (2011) 174–215.
  • [8] J. G. Kemeny, J. L. Snell and A. W. Knapp. Denumerable Markov Chains. van Nostrand, Princeton, NJ, 1966.
  • [9] D. Kious and V. Sidoravicius. Phase transition for the once-reinforced random walk on $\mathbb{Z}_{d}$-like trees. Available at arXiv:1604.07631.
    arXiv: 1604.07631
  • [10] R. Lyons and R. Pemantle. Random walk in a random environment and first-passage percolation on trees. Ann. Probab. 20 (1992) 125–136.
  • [11] M. V. Menshikov and D. Petritis. On random walks in random environment on trees and their relationship with multiplicative chaos. In Mathematics and Computer Science II (Versailles, 2002) 415–422. Birkhäuser, Basel, 2002.
  • [12] R. Pemantle. A survey of random processes with reinforcement. Probab. Surv. 4 (2007) 1–79.
  • [13] A. Sznitman. Topics in random walks in random environment. In School and Conference in Probability Theory 203–266 (electronic). ICTP Lect. Notes XVII. Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004. Available at http://users.ictp.trieste.it/~pub_off/lectures/lns017/Sznitman/Sznitman.pdf.

Institut Henri Poincaré

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more