Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

One-dimensional finite range random walk in random medium and invariant measure equation

Julien Brémont

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Abstract

We consider a model of random walks on ℤ with finite range in a stationary and ergodic random environment. We first provide a fine analysis of the geometrical properties of the central left and right Lyapunov eigenvectors of the random matrix naturally associated with the random walk, highlighting the mechanism of the model. This allows us to formulate a criterion for the existence of the absolutely continuous invariant measure for the environments seen from the particle. We then deduce a characterization of the non-zero-speed regime of the model.

Résumé

Nous considérons un modèle de marche aléatoire sur ℤ à pas bornés en environnement aléatoire stationnaire ergodique. Dans une première partie, nous détaillons les propriétés géométriques des vecteurs propres de Lyapunov centraux pour la matrice aléatoire naturellement associée à la marche, mettant en lumière le mécanisme du modèle. Nous formulons alors un critère, vectoriel dans les situations transientes, pour l’existence de la mesure invariante absolument continue pour les environnements vus depuis la particule. En corollaire, nous obtenons une caractérisation du régime avec vitesse non nulle.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 1 (2009), 70-103.

Dates
First available in Project Euclid: 12 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1234469972

Digital Object Identifier
doi:10.1214/07-AIHP150

Mathematical Reviews number (MathSciNet)
MR2500229

Zentralblatt MATH identifier
1171.60395

Subjects
Primary: 60F15: Strong theorems 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K37: Processes in random environments

Keywords
Finite range Markov chain Lyapunov eigenvector Invariant measure Stable cone

Citation

Brémont, Julien. One-dimensional finite range random walk in random medium and invariant measure equation. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 1, 70--103. doi:10.1214/07-AIHP150. https://projecteuclid.org/euclid.aihp/1234469972


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