Process-level quenched large deviations for random walk in random environment
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 214-242.

Nous considérons une marche aléatoire en environnement aléatoire ergodique. La marche est elliptique et à pas bornés. Nous prouvons un principe de grandes déviations au niveau 3, sous presque tout environnement, avec une fonctionnelle d'action liée à une entropie relative.

We consider a bounded step size random walk in an ergodic random environment with some ellipticity, on an integer lattice of arbitrary dimension. We prove a level 3 large deviation principle, under almost every environment, with rate function related to a relative entropy.

DOI : 10.1214/10-AIHP369
Classification : 60K37, 60F10, 82D30, 82C44
Mots-clés : random walk, random environment, RWRE, large deviation, environment process, relative entropy, homogenization 
@article{AIHPB_2011__47_1_214_0,
     author = {Rassoul-Agha, Firas and Sepp\"al\"ainen, Timo},
     title = {Process-level quenched large deviations for random walk in random environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {214--242},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {1},
     year = {2011},
     doi = {10.1214/10-AIHP369},
     mrnumber = {2779403},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/10-AIHP369/}
}
TY  - JOUR
AU  - Rassoul-Agha, Firas
AU  - Seppäläinen, Timo
TI  - Process-level quenched large deviations for random walk in random environment
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2011
SP  - 214
EP  - 242
VL  - 47
IS  - 1
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/10-AIHP369/
DO  - 10.1214/10-AIHP369
LA  - en
ID  - AIHPB_2011__47_1_214_0
ER  - 
%0 Journal Article
%A Rassoul-Agha, Firas
%A Seppäläinen, Timo
%T Process-level quenched large deviations for random walk in random environment
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2011
%P 214-242
%V 47
%N 1
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/10-AIHP369/
%R 10.1214/10-AIHP369
%G en
%F AIHPB_2011__47_1_214_0
Rassoul-Agha, Firas; Seppäläinen, Timo. Process-level quenched large deviations for random walk in random environment. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 214-242. doi : 10.1214/10-AIHP369. http://archive.numdam.org/articles/10.1214/10-AIHP369/

[1] C. Boldrighini, R. A. Minlos and A. Pellegrinotti. Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive. Probab. Theory Related Fields 129 (2004) 133-156. | MR | Zbl

[2] F. Comets, N. Gantert and O. Zeitouni. Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probab. Theory Related Fields 118 (2000) 65-114. | MR | Zbl

[3] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Applications of Mathematics 38. Springer, New York, 1998. | MR | Zbl

[4] F. Den Hollander. Large Deviations. Fields Institute Monographs 14. Amer. Math. Soc., Providence, RI, 2000. | MR | Zbl

[5] J.-D. Deuschel and D. W. Stroock. Large Deviations. Pure and Applied Mathematics 137. Academic Press, Boston, MA, 1989. | MR | Zbl

[6] M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time. I. Comm. Pure Appl. Math. 28 (1975) 1-47. | MR | Zbl

[7] M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time. III. Comm. Pure Appl. Math. 29 (1976) 389-461. | MR | Zbl

[8] I. Ekeland and R. Témam. Convex Analysis and Variational Problems, English edition. Classics in Applied Mathematics 28. SIAM, Philadelphia, PA, 1999. | MR | Zbl

[9] H.-O. Georgii. Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics 9. Walter de Gruyter, Berlin, 1988. | MR | Zbl

[10] A. Greven and F. Den Hollander. Large deviations for a random walk in random environment. Ann. Probab. 22 (1994) 1381-1428. | MR | Zbl

[11] G. Kassay. A simple proof for König's minimax theorem. Acta Math. Hungar. 63 (1994) 371-374. | MR | Zbl

[12] E. Kosygina, F. Rezakhanlou and S. R. S. Varadhan. Stochastic homogenization of Hamilton-Jacobi-Bellman equations. Comm. Pure Appl. Math. 59 (2006) 1489-1521. | MR | Zbl

[13] 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

[14] F. Rassoul-Agha and T. Seppäläinen. An almost sure invariance principle for random walks in a space-time random environment. Probab. Theory Related Fields 133 (2005) 299-314. | MR | Zbl

[15] F. Rassoul-Agha and T. Seppäläinen. A course on large deviation theory with an introduction to Gibbs measures. Preprint, 2009.

[16] M. Rosenblatt. Markov Processes. Structure and Asymptotic Behavior. Springer, New York, 1971. | MR | Zbl

[17] J. Rosenbluth. Quenched large deviations for multidimensional random walk in random environment: A variational formula. Thesis dissertation, New York University, 2006. Available at http://arxiv.org/abs/0804.1444. | MR

[18] W. Rudin. Functional Analysis, 2nd edition. McGraw-Hill, New York, 1991. | MR | Zbl

[19] T. Seppäläinen. Large deviations for lattice systems. I. Parametrized independent fields. Probab. Theory Related Fields 96 (1993) 241-260. | MR | Zbl

[20] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Springer, Berlin, 2006. | MR | Zbl

[21] S. R. S. Varadhan. Large Deviations and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics 46. SIAM, Philadelphia, PA, 1984. | MR | Zbl

[22] S. R. S. Varadhan. Large deviations for random walks in a random environment. Comm. Pure Appl. Math. 56 (2003) 1222-1245. Dedicated to the memory of Jürgen K. Moser. | MR | Zbl

[23] A. Yilmaz. Large deviations for random walk in a space-time product environment. Ann. Probab. 37 (2009a) 189-205. | MR | Zbl

[24] A. Yilmaz. Quenched large deviations for random walk in a random environment. Comm. Pure Appl. Math. 62 (2009b) 1033-1075. | MR | Zbl

[25] M. P. W. Zerner. Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment. Ann. Probab. 26 (1998) 1446-1476. | MR | Zbl

Cité par Sources :