Dans son article de 2003, Varadhan démontre un principe de grandes déviations pour la loi moyennée de la vitesse d'une particule suivant une marche aléatoire au plus proche voisin dans un environnement i.i.d. elliptique sur ℤd avec d≥1, et donne une formule variationnelle pour la fonction de taux correspondante Ia. Sous la condition (T) de transience de Sznitman, nous montrons que Ia est strictement convexe et analytique dans un ouvert non vide , et que la vraie vitesse de la particule est un élément de (resp. un élément de la frontière de ) quand la marche est “non-nichée” (resp. nichée). Nous identifions alors l'unique minimisant de la formule variationnelle de Varadhan pour toute vélocité de .
In his 2003 paper, Varadhan proves the averaged large deviation principle for the mean velocity of a particle taking a nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on ℤd with d≥1, and gives a variational formula for the corresponding rate function Ia. Under Sznitman's transience condition (T), we show that Ia is strictly convex and analytic on a non-empty open set , and that the true velocity of the particle is an element (resp. in the boundary) of when the walk is non-nestling (resp. nestling). We then identify the unique minimizer of Varadhan's variational formula at any velocity in .
Mots clés : disordered media, rare events, rate function, regeneration times
@article{AIHPB_2010__46_3_853_0, author = {Yilmaz, Atilla}, title = {Averaged large deviations for random walk in a random environment}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {853--868}, publisher = {Gauthier-Villars}, volume = {46}, number = {3}, year = {2010}, doi = {10.1214/09-AIHP332}, mrnumber = {2682269}, zbl = {1201.60098}, language = {en}, url = {http://www.numdam.org/articles/10.1214/09-AIHP332/} }
TY - JOUR AU - Yilmaz, Atilla TI - Averaged large deviations for random walk in a random environment JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2010 SP - 853 EP - 868 VL - 46 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/09-AIHP332/ DO - 10.1214/09-AIHP332 LA - en ID - AIHPB_2010__46_3_853_0 ER -
%0 Journal Article %A Yilmaz, Atilla %T Averaged large deviations for random walk in a random environment %J Annales de l'I.H.P. Probabilités et statistiques %D 2010 %P 853-868 %V 46 %N 3 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/09-AIHP332/ %R 10.1214/09-AIHP332 %G en %F AIHPB_2010__46_3_853_0
Yilmaz, Atilla. Averaged large deviations for random walk in a random environment. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 853-868. doi : 10.1214/09-AIHP332. http://www.numdam.org/articles/10.1214/09-AIHP332/
[1] Limiting velocity of high-dimensional random walk in random environment. Ann. Probab. 36 (2008) 728-738. | MR | Zbl
.[2] Quenched, annealed and functional large deviations for one dimensional random walk in random environment. Probab. Theory Related Fields 118 (2000) 65-114. | MR | Zbl
, and .[3] An invariance principle for reversible Markov processes with applications to random motions in random environments. J. Stat. Phys. 55 (1989) 787-855. | MR | Zbl
, , and .[4] Large Deviation Techniques and Applications, 2nd edition. Springer, New York, 1998. | MR | Zbl
and .[5] Asymptotic evaluation of certain Markov process expectations for large time. IV. Comm. Pure Appl. Math. 36 (1983) 183-212. | MR | Zbl
and .[6] Large deviations for a random walk in random environment. Ann. Probab. 22 (1994) 1381-1428. | MR | Zbl
and .[7] A central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion. Comm. Math. Phys. 104 (1986) 1-19. | MR | Zbl
and .[8] The averaging method and walks in inhomogeneous environments. Russian Math. Surveys (Uspekhi Mat. Nauk) 40 (1985) 73-145. | MR | Zbl
.[9] The Implicit Function Theorem: History, Theory, and Applications. Birkhäuser, Boston, 2002. | MR | Zbl
and .[10] Homogenization of Diffusion Processes in Random Fields. Ecole Polytecnique, Palaiseau, 1994.
.[11] Boundary value problems with rapidly oscillating random coefficients. In Random Fields. J. Fritz and D. Szasz (eds). Janyos Bolyai Series. North-Holland, Amsterdam, 1981. | MR | Zbl
and .[12] Limiting distributions and large deviations for random walks in random environments. Ph.D. thesis, Univ. Minnesota, 2008. | MR
.[13] On the annealed large deviation rate function for a multi-dimensional random walk in random environment. ALEA. To appear. Preprint, 2008. Available at arXiv:0812.3619. | MR
and .[14] Large deviations for random walks in a mixing random environment and other (non-Markov) random walks. Comm. Pure Appl. Math. 57 (2004) 1178-1196. | MR | Zbl
.[15] Quenched large deviations for multidimensional random walk in random environment: A variational formula. Ph.D. thesis, Courant Institute, New York Univ., 2006. Available at arXiv:0804.1444. | MR
.[16] A law of large numbers for random walks in random environment. Ann. Probab. 27 (1999) 1851-1869. | MR | Zbl
and .[17] Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. 2 (2000) 93-143. | MR | Zbl
.[18] On a class of transient random walks in random environment. Ann. Probab. 29 (2001) 724-765. | MR | Zbl
.[19] Lectures on random motions in random media. In Ten Lectures on Random Media. DMV-Lectures 32. Birkhäuser, Basel, 2002. | MR | Zbl
.[20] Large deviations for random walks in a random environment. Comm. Pure Appl. Math. 56 (2003) 1222-1245. | MR | Zbl
.[21] Large deviations for random walk in a random environment. Ph.D. thesis, Courant Institute, New York Univ., 2008. Available at arXiv:0809.1227. | MR
.[22] Quenched large deviations for random walk in a random environment. Comm. Pure Appl. Math. 62 (2009) 1033-1075. | MR | Zbl
.[23] On the equality of the quenched and averaged large deviation rate functions for high-dimensional ballistic random walk in a random environment. Preprint, 2009. Available at arXiv:0903.0410. | MR
.[24] Random walks in random environments. J. Phys. A: Math. Gen. 39 (2006) R433-R464. | MR | Zbl
.[25] Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment. Ann. Probab. 26 (1998) 1446-1476. | MR | Zbl
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