# Meeting time of independent random walks in random environment

ESAIM: Probability and Statistics (2013)

- Volume: 17, page 257-292
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topGallesco, Christophe. "Meeting time of independent random walks in random environment." ESAIM: Probability and Statistics 17 (2013): 257-292. <http://eudml.org/doc/273618>.

@article{Gallesco2013,

abstract = {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.},

author = {Gallesco, Christophe},

journal = {ESAIM: Probability and Statistics},

keywords = {random walk in random environment; Sinai’s regime; t-stable point; meeting time; coalescing time; Sinai's regime; t-stable pint},

language = {eng},

pages = {257-292},

publisher = {EDP-Sciences},

title = {Meeting time of independent random walks in random environment},

url = {http://eudml.org/doc/273618},

volume = {17},

year = {2013},

}

TY - JOUR

AU - Gallesco, Christophe

TI - Meeting time of independent random walks in random environment

JO - ESAIM: Probability and Statistics

PY - 2013

PB - EDP-Sciences

VL - 17

SP - 257

EP - 292

AB - 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.

LA - eng

KW - random walk in random environment; Sinai’s regime; t-stable point; meeting time; coalescing time; Sinai's regime; t-stable pint

UR - http://eudml.org/doc/273618

ER -

## References

top- [1] V. Belitsky, P. Ferrari, M. Menshikov and S. Popov, A mixture of the exclusion process and the voter model. Bernoulli7 (2001) 119–144. Zbl0978.60105MR1811747
- [2] W. Böhm and S.G. Mohanty, On the Karlin–McGregor theorem and applications. Ann. Appl. Probab.7 (1997) 314–325. Zbl0884.60010MR1442315
- [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. Fields126 (2003) 571–609. Zbl1027.60091MR2001198
- [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. Zbl1171.60396MR2085605
- [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. Zbl0938.60065MR1675023
- [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. Fields137 (2007) 443–473. Zbl1106.60082MR2278464
- [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. Zbl1186.60108MR2574090
- [8] A. Fribergh, N. Gantert and S.Yu. Popov, On slowdown and speedup of transient random walks in random environment. Probab. Theory Relat. Fields147 (2010) 43–88. Zbl1193.60122MR2594347
- [9] C. Gallesco, On the moments of the meeting time of independent random walks in random environment. arXiv:0903.4697 (2009). Zbl1292.60098MR3021319
- [10] N. Gantert, Y. Peres and Z. Shi, The infinite valley for a recurrent random walk in random environment. Ann. Inst. Henri Poincaré46 (2010) 525–536. Zbl1201.60096MR2667708
- [11] A. Greven and F. den Hollander, Large deviations for a random walk in random environment. Ann. Probab. 22 (1994) 1381 − 1428. Zbl0820.60054MR1303649
- [12] Y. Hu and Z. Shi, Moderate deviations for diffusions with Brownian potentials. Ann. Probab.32 (2004) 3191–3220. Zbl1066.60096MR2094443
- [13] B. Hughes, Random Walks and Random Environments. The Clarendon Press, Oxford University Press, New York. Random Environments 2 (1996). Zbl0925.60076
- [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. Zbl0388.60069MR380998
- [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. Gebiete32 (1975) 111–131. Zbl0308.60029MR375412
- [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. Zbl0885.60061MR1490046
- [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). Zbl1031.60088MR2226845
- [18] Ya.G. Sinai, The limiting behavior of one-dimensional random walk in random medium. Theory Probab. Appl.27 (1982) 256–268. Zbl0505.60086MR657919
- [19] F. Solomon, Random walks in a random environment. Ann. Probab.3 (1975) 1–31. Zbl0305.60029MR362503
- [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. Zbl1060.60103MR2071631

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.