Meeting time of independent random walks in random environment
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 257-292.

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.

DOI : 10.1051/ps/2011159
Classification : 60K37
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://archive.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://archive.numdam.org/articles/10.1051/ps/2011159/
DO  - 10.1051/ps/2011159
LA  - en
ID  - PS_2013__17__257_0
ER  - 
%0 Journal Article
%A Gallesco, Christophe
%T Meeting time of independent random walks in random environment
%J ESAIM: Probability and Statistics
%D 2013
%P 257-292
%V 17
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2011159/
%R 10.1051/ps/2011159
%G en
%F PS_2013__17__257_0
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://archive.numdam.org/articles/10.1051/ps/2011159/

[1] V. Belitsky, P. Ferrari, M. Menshikov and S. Popov, A mixture of the exclusion process and the voter model. Bernoulli 7 (2001) 119-144. | MR | Zbl

[2] W. Böhm and S.G. Mohanty, On the Karlin-McGregor theorem and applications. Ann. Appl. Probab. 7 (1997) 314-325. | MR | Zbl

[3] F. Comets and S.Yu. Popov, 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] F. Comets and S.Yu. Popov, A note on quenched moderate deviations for Sinai's random walk in random environment. ESAIM : PS 8 (2004) 56-65. | EuDML | MR | Zbl

[5] F. Comets, M.V. Menshikov and S.Yu. Popov, Lyapunov functions for random walks and strings in random environment. Ann. Probab. 26 (1998) 1433-1445. | MR | Zbl

[6] A. Dembo, N. Gantert, Y. Peres and Z. Shi, Valleys and the maximal local time for random walk in random environment. Probab. Theory Relat. Fields 137 (2007) 443-473. | MR | Zbl

[7] N. Enriquez, C. Sabot and O. Zindy, 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] A. Fribergh, N. Gantert and S.Yu. Popov, On slowdown and speedup of transient random walks in random environment. Probab. Theory Relat. Fields 147 (2010) 43-88. | MR | Zbl

[9] C. Gallesco, On the moments of the meeting time of independent random walks in random environment. arXiv:0903.4697 (2009).

[10] N. Gantert, Y. Peres and Z. Shi, The infinite valley for a recurrent random walk in random environment. Ann. Inst. Henri Poincaré 46 (2010) 525-536. | EuDML | Numdam | MR | Zbl

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

[12] Y. Hu and Z. Shi, Moderate deviations for diffusions with Brownian potentials. Ann. Probab. 32 (2004) 3191-3220. | MR | Zbl

[13] B. Hughes, Random Walks and Random Environments. The Clarendon Press, Oxford University Press, New York. Random Environments 2 (1996). | MR | Zbl

[14] H. Kesten, M.V. Kozlov and F. Spitzer, A limit law for random walk in a random environment. Compos. Math. 30 (1975) 145-168. | EuDML | Numdam | MR | Zbl

[15] J. Komlós, P. Major and G. Tusnády, 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] L. Saloff-Coste, 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] Z. Shi, 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] Ya.G. Sinai, The limiting behavior of one-dimensional random walk in random medium. Theory Probab. Appl. 27 (1982) 256-268. | MR | Zbl

[19] F. Solomon, Random walks in a random environment. Ann. Probab. 3 (1975) 1-31. | MR | Zbl

[20] O. Zeitouni, 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 :