One-dimensional finite range random walk in random medium and invariant measure equation
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, p. 70-103

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.

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.

DOI : https://doi.org/10.1214/07-AIHP150
Classification:  60F15,  60J10,  60K37
Keywords: finite range Markov chain, Lyapunov eigenvector, invariant measure, stable cone
@article{AIHPB_2009__45_1_70_0,
     author = {Br\'emont, Julien},
     title = {One-dimensional finite range random walk in random medium and invariant measure equation},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     pages = {70-103},
     doi = {10.1214/07-AIHP150},
     zbl = {1171.60395},
     mrnumber = {2500229},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_1_70_0}
}
Brémont, Julien. One-dimensional finite range random walk in random medium and invariant measure equation. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, pp. 70-103. doi : 10.1214/07-AIHP150. http://www.numdam.org/item/AIHPB_2009__45_1_70_0/

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