We consider, in the continuous time version, γ independent random walks on Z+ in random environment in Sinai's regime. Let Tγ be the first meeting time of one pair of the γ random walks starting at different positions. We first show that the tail of the quenched distribution of Tγ, after a suitable rescaling, converges in probability, to some functional of the Brownian motion. Then we compute the law of this functional. Eventually, we obtain results about the moments of this meeting time. Being Eω the quenched expectation, we show that, for almost all environments ω, Eω[Tγc] is finite for c < γ(γ - 1) / 2 and infinite for c > γ(γ - 1) / 2.
Mots-clés : random walk in random environment, Sinai's regime, t-stable point, meeting time, coalescing time
@article{PS_2013__17__257_0,
author = {Gallesco, Christophe},
title = {Meeting time of independent random walks in random environment},
journal = {ESAIM: Probability and Statistics},
pages = {257--292},
publisher = {EDP-Sciences},
volume = {17},
year = {2013},
doi = {10.1051/ps/2011159},
mrnumber = {3021319},
zbl = {1292.60098},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2011159/}
}
TY - JOUR AU - Gallesco, Christophe TI - Meeting time of independent random walks in random environment JO - ESAIM: Probability and Statistics PY - 2013 SP - 257 EP - 292 VL - 17 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2011159/ DO - 10.1051/ps/2011159 LA - en ID - PS_2013__17__257_0 ER -
Gallesco, Christophe. Meeting time of independent random walks in random environment. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 257-292. doi : 10.1051/ps/2011159. http://www.numdam.org/articles/10.1051/ps/2011159/
[1] , , and , A mixture of the exclusion process and the voter model. Bernoulli 7 (2001) 119-144. | MR | Zbl
[2] and , On the Karlin-McGregor theorem and applications. Ann. Appl. Probab. 7 (1997) 314-325. | MR | Zbl
[3] and , Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment. Probab. Theory Relat. Fields 126 (2003) 571-609. | MR | Zbl
[4] and , A note on quenched moderate deviations for Sinai's random walk in random environment. ESAIM : PS 8 (2004) 56-65. | EuDML | MR | Zbl
[5] , and , Lyapunov functions for random walks and strings in random environment. Ann. Probab. 26 (1998) 1433-1445. | MR | Zbl
[6] , , and , Valleys and the maximal local time for random walk in random environment. Probab. Theory Relat. Fields 137 (2007) 443-473. | MR | Zbl
[7] , and , Aging and quenched localization one-dimensional random walks in random environment in the bub-ballistic regime. Bulletin de la S.M.F. 137 (2009) 423-452. | EuDML | Numdam | MR | Zbl
[8] , and , On slowdown and speedup of transient random walks in random environment. Probab. Theory Relat. Fields 147 (2010) 43-88. | MR | Zbl
[9] , On the moments of the meeting time of independent random walks in random environment. arXiv:0903.4697 (2009).
[10] , and , The infinite valley for a recurrent random walk in random environment. Ann. Inst. Henri Poincaré 46 (2010) 525-536. | EuDML | Numdam | MR | Zbl
[11] and , Large deviations for a random walk in random environment. Ann. Probab. 22 (1994) 1381 − 1428. | MR | Zbl
[12] and , Moderate deviations for diffusions with Brownian potentials. Ann. Probab. 32 (2004) 3191-3220. | MR | Zbl
[13] , Random Walks and Random Environments. The Clarendon Press, Oxford University Press, New York. Random Environments 2 (1996). | MR | Zbl
[14] , and , A limit law for random walk in a random environment. Compos. Math. 30 (1975) 145-168. | EuDML | Numdam | MR | Zbl
[15] , and , An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitstheor. Verw. Gebiete 32 (1975) 111-131. | MR | Zbl
[16] , Lectures on Finite Markov Chains. Lectures on probability theory and statistics, Saint-Flour, 1996, Springer, Berlin. Lect. Notes Math. 1665 (1997) 301-413. | MR | Zbl
[17] , Sinai's Walk via Stochastic Calculus, in Milieux Aléatoires Panoramas et Synthèses 12, edited by F. Comets and E. Pardoux. Société Mathématique de France, Paris (2001). | MR | Zbl
[18] , The limiting behavior of one-dimensional random walk in random medium. Theory Probab. Appl. 27 (1982) 256-268. | MR | Zbl
[19] , Random walks in a random environment. Ann. Probab. 3 (1975) 1-31. | MR | Zbl
[20] , Lecture Notes on Random Walks in Random Environment given at the 31st Probability Summer School in Saint-Flour, Springer. Lect. Notes Math. 1837 (2004) 191-312. | MR | Zbl
Cité par Sources :
