Meeting time of independent random walks in random environment

Christophe Gallesco

ESAIM: Probability and Statistics (2013)

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

Abstract

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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.

How to cite

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Gallesco, 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 &lt; γ(γ − 1) / 2 and infinite for c &gt; 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 &lt; γ(γ − 1) / 2 and infinite for c &gt; 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

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