Nous considérons une marche aléatoire multidimensionnelle en environnement aléatoire produit. La marche est à pas bornés, transiente dans une direction spatiale donnée, et telle que le temps de régénération posséde un moment suffisamment haut. Nous prouvons un principe d'invariance, ou un théorème limite central fonctionnel, sous presque tout environnement pour la marche centrée et diffusivement normalisée. Le point principal derrière le principe d'invariance est que la moyenne trempée (quenched) de la marche est sous-diffusive.
We consider a multidimensional random walk in a product random environment with bounded steps, transience in some spatial direction, and high enough moments on the regeneration time. We prove an invariance principle, or functional central limit theorem, under almost every environment for the diffusively scaled centered walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively.
Mots-clés : random walk, ballistic, random environment, central limit theorem, invariance principle, point of view of the particle, environment process, Green function
@article{AIHPB_2009__45_2_373_0,
author = {Rassoul-Agha, Firas and Sepp\"al\"ainen, Timo},
title = {Almost sure functional central limit theorem for ballistic random walk in random environment},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
pages = {373--420},
publisher = {Gauthier-Villars},
volume = {45},
number = {2},
year = {2009},
doi = {10.1214/08-AIHP167},
mrnumber = {2521407},
zbl = {1176.60087},
language = {en},
url = {http://archive.numdam.org/articles/10.1214/08-AIHP167/}
}
TY - JOUR AU - Rassoul-Agha, Firas AU - Seppäläinen, Timo TI - Almost sure functional central limit theorem for ballistic random walk in random environment JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2009 SP - 373 EP - 420 VL - 45 IS - 2 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/08-AIHP167/ DO - 10.1214/08-AIHP167 LA - en ID - AIHPB_2009__45_2_373_0 ER -
%0 Journal Article %A Rassoul-Agha, Firas %A Seppäläinen, Timo %T Almost sure functional central limit theorem for ballistic random walk in random environment %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 373-420 %V 45 %N 2 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/08-AIHP167/ %R 10.1214/08-AIHP167 %G en %F AIHPB_2009__45_2_373_0
Rassoul-Agha, Firas; Seppäläinen, Timo. Almost sure functional central limit theorem for ballistic random walk in random environment. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 2, pp. 373-420. doi : 10.1214/08-AIHP167. http://archive.numdam.org/articles/10.1214/08-AIHP167/
[1] and . A quenched invariance principle for certain ballistic random walks in i.i.d. environments, 2008. Available at http://front.math.ucdavis.edu/math.PR/0702306. | MR | Zbl
[2] and . On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9 (2002) 345-375. Special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday. | MR | Zbl
[3] and . Ten Lectures on Random Media. Birkhäuser, Basel, 2002. | MR | Zbl
[4] and . Random walks in asymmetric random environments. Comm. Math. Phys. 142 (1991) 345-420. | MR | Zbl
[5] . Distribution function inequalities for martingales. Ann. Probability 1 (1973) 19-42. | MR | Zbl
[6] and . The central limit theorem for Markov chains started at a point. Probab. Theory Related Fields 125 (2003) 73-76. | MR | Zbl
[7] . Probability: Theory and Examples, 3rd edition. Brooks/Cole-Thomson, Belmont, CA, 2004. | MR | Zbl
[8] and . Markov Processes. Wiley, New York, 1986. | MR | Zbl
[9] . An Introduction to Probability Theory and Its Applications. Vol. II, 2nd edition. Wiley, New York, 1971. | MR | Zbl
[10] . Simple transient random walks in one-dimensional random environment: the central limit theorem. Probab. Theory Related Fields 139 (2007) 41-64. | MR | Zbl
[11] and . Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2000) 713-724. | MR | Zbl
[12] and . An almost sure invariance principle for random walks in a space-time random environment. Probab. Theory Related Fields 133 (2005) 299-314. | MR | Zbl
[13] and . Ballistic random walk in a random environment with a forbidden direction. ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 111-147 (electronic). | MR | Zbl
[14] and . An almost sure invariance principle for ballistic random walks in product random environment, 2007. Available at http://front.math.ucdavis.edu/math.PR/0704.1022.
[15] and . Quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction. Ann. Probab. 35 (2007) 1-31. | MR | Zbl
[16] . Markov Processes. Structure and Asymptotic Behavior. Springer, New York, 1971. | MR | Zbl
[17] . Principles of Random Walks, 2nd edition. Springer, New York, 1976. | MR | Zbl
[18] . Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. (JEMS) 2 (2000) 93-143. | MR | Zbl
[19] . An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 122 (2002) 509-544. | MR | Zbl
[20] . Topics in random walks in random environment. In School and Conference on Probability Theory. ICTP Lect. Notes, XVII. Abdus Salam Int. Cent. Theoret. Phys., Trieste (2004) 203-266 (electronic). | MR | Zbl
[21] and . An invariance principle for isotropic diffusions in random environment. Invent. Math. 164 (2006) 455-567. | MR | Zbl
[22] and . A law of large numbers for random walks in random environment. Ann. Probab. 27 (1999) 1851-1869. | MR | Zbl
[23] . Random Walks in Random Environments. Springer, Berlin, 2004. | Zbl
[24] . 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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