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

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.

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.

DOI : 10.1214/07-AIHP150
Classification : 60F15, 60J10, 60K37
Mots-clés : 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},
     pages = {70--103},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     doi = {10.1214/07-AIHP150},
     mrnumber = {2500229},
     zbl = {1171.60395},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/07-AIHP150/}
}
TY  - JOUR
AU  - Brémont, Julien
TI  - One-dimensional finite range random walk in random medium and invariant measure equation
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 70
EP  - 103
VL  - 45
IS  - 1
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/07-AIHP150/
DO  - 10.1214/07-AIHP150
LA  - en
ID  - AIHPB_2009__45_1_70_0
ER  - 
%0 Journal Article
%A Brémont, Julien
%T One-dimensional finite range random walk in random medium and invariant measure equation
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 70-103
%V 45
%N 1
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/07-AIHP150/
%R 10.1214/07-AIHP150
%G en
%F 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, Tome 45 (2009) no. 1, pp. 70-103. doi : 10.1214/07-AIHP150. http://archive.numdam.org/articles/10.1214/07-AIHP150/

[1] S. Alili. Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36 (1999) 334-349. | MR | Zbl

[2] L. Arnold. Random Dynamical Systems. Springer, Berlin, 1998. | MR | Zbl

[3] E. Bolthausen and I. Goldsheid. Recurrence and transience of random walks in random environments on a strip. Comm. Math. Phys. 214 (2000) 429-447. | MR | Zbl

[4] E. Bolthausen and I. Goldsheid. Lingering random walks in random environment on a strip. Comm. Math. Phys. 278 (2008) 253-288. | MR | Zbl

[5] J. Brémont. On the recurrence of random walks on ℤ in random medium. C. R. Acad. Sci. Paris Sér. I Math. 333 (2001) 1011-1016. | MR | Zbl

[6] J. Brémont. On some random walks on ℤ in random medium. Ann. Probab. 30 (2002) 1266-1312. | MR | Zbl

[7] J. Brémont. Random walks on ℤ in random medium and Lyapunov spectrum. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004) 309-336. | EuDML | Numdam | MR | Zbl

[8] J.-P. Conze and Y. Guivarc'H. Marches en milieu aléatoire et mesures quasi-invariantes pour un système dynamique. Colloq. Math. 84/85 (2000) 457-480. (Dedicated to the memory of Anzelm Iwanik.) | EuDML | MR | Zbl

[9] C. Evstigneev. Positive matrix-valued cocycles over dynamical systems. Uspekhi Mat. Nauk 29 (1974) 219-226. | MR | Zbl

[10] H. Federer. Geometric Measure Theory. Springer, New York, 1969. | MR | Zbl

[11] I. Goldsheid. Linear and sublinear growth and the CLT for hitting times of a random walk in random environment on a strip. Probab. Theory Related Fields 141 (2008) 471-511. | MR | Zbl

[12] H. Hennion. Limit theorems for products of positive random matrices. Ann. Probab. 25 (1997) 1545-1587. | MR | Zbl

[13] M. Karoubi and C. Leruste. Algebraic Topology via Differential Geometry. Cambridge Univ. Press, 1987. | MR | Zbl

[14] E. Key. Recurrence and transience criteria for a random walk in a random environment. Ann. Probab. 12 (1984) 529-560. | MR | Zbl

[15] S. Kozlov. The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40 (1985) 61-120, 238. | MR | Zbl

[16] F. Ledrappier. Quelques propriétés des exposants caractéristiques. Ecole d'été de Saint-Flour 1982 305-396. Lecture Notes in Math. 1097. Springer, Berlin, 1984. | MR | Zbl

[17] A. Lëtchikov. Localization of One-dimensional Random Walks in Random Environments. Harwood Academic Publishers, Chur, 1989. | MR

[18] A. Lëtchikov. A criterion for the applicability of the central limit theorem to one-dimensional random walks in random environments. Teor. Veroyatnost. i Primenen. 37 (1992) 576-580. | MR | Zbl

[19] A. Lëtchikov. A criterion for linear drift, and the central limit theorem for one-dimensional random walks in random environments. Russian Acad. Sci. Sb. Math. 79 (1994) 73-92. | MR | Zbl

[20] V. Oseledec. A multiplicative ergodic theorem. Characteristic Ljapunov exponents of dynamical systems. Trudy Moskov. Mat. Obšč. 19 (1968) 179-210. | MR | Zbl

[21] Y. Peres. Domains of analytic continuation for the top Lyapunov exponent. Ann. Inst. H. Poincaré 28 (1992) 131-148. | Numdam | MR | Zbl

[22] F. Rassoul-Agha. The point of view of the particle on the Law of Large Numbers for random walks in a mixing random environment. Ann. Probab. 31 (2003) 1441-1463. | MR | Zbl

[23] A. Raugi. Théorème ergodique multiplicatif. Produits de matrices aléatoires indépendantes. Fascicule de probabilités, Univ. Rennes I, 1997. | Zbl

[24] A. Roitershtein. Transient random walks on a strip in a random environment. Ann. Probab. 36 (2008) 2354-2387. | MR | Zbl

[25] A.-S. Sznitman. Topics in random walks in random environment. Lecture given at the School and Conference on Probability Theory, Trieste, May 2002. ICTP Lect. Notes XVII, 2004. | MR | Zbl

[26] O. Zeitouni. Random walks in random environment. Lectures on probability theory and statistics. Ecole d'Eté de probabilités de Saint-Flour XXXI - 2001 191-312. Lecture Notes in Math. 1837. Springer, Berlin, 2004. | MR | Zbl

Cité par Sources :