The infinite valley for a recurrent random walk in random environment

Nina Gantert; Yuval Peres; Zhan Shi

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 2, page 525-536
  • ISSN: 0246-0203

Abstract

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We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the – suitably centered – empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see Comm. Math. Phys.92 (1984) 491–506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.

How to cite

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Gantert, Nina, Peres, Yuval, and Shi, Zhan. "The infinite valley for a recurrent random walk in random environment." Annales de l'I.H.P. Probabilités et statistiques 46.2 (2010): 525-536. <http://eudml.org/doc/239998>.

@article{Gantert2010,
abstract = {We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the – suitably centered – empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see Comm. Math. Phys.92 (1984) 491–506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.},
author = {Gantert, Nina, Peres, Yuval, Shi, Zhan},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random walk in random environment; empirical distribution; local time; self-intersection local time},
language = {eng},
number = {2},
pages = {525-536},
publisher = {Gauthier-Villars},
title = {The infinite valley for a recurrent random walk in random environment},
url = {http://eudml.org/doc/239998},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Gantert, Nina
AU - Peres, Yuval
AU - Shi, Zhan
TI - The infinite valley for a recurrent random walk in random environment
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 2
SP - 525
EP - 536
AB - We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the – suitably centered – empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see Comm. Math. Phys.92 (1984) 491–506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.
LA - eng
KW - random walk in random environment; empirical distribution; local time; self-intersection local time
UR - http://eudml.org/doc/239998
ER -

References

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  1. [1] J. Bertoin. Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stochastic Process. Appl. 47 (1993) 17–35. Zbl0786.60101MR1232850
  2. [2] R. Billing. Häufig besuchte Punkte der Irrfahrt in zufälliger Umgebung. Diploma thesis, Johannes Gutenberg-Universität Mainz, 2009. 
  3. [3] A. Dembo, N. Gantert, Y. Peres and Z. Shi. Valleys and the maximal local time for random walk in random environment. Probab. Theory Related Fields 137 (2007) 443–473. Zbl1106.60082MR2278464
  4. [4] P. G. Doyle and E. J. Snell. Probability: Random Walks and Electrical Networks. Carus Math. Monographs 22. Math. Assoc. Amer., Washington, DC, 1984. Zbl0583.60065MR920811
  5. [5] N. Gantert and Z. Shi. Many visits to a single site by a transient random walk in random environment. Stochastic Process. Appl. 99 (2002) 159–176. Zbl1059.60100MR1901151
  6. [6] A. O. Golosov. Localization of random walks in one-dimensional random environments. Comm. Math. Phys. 92 (1984) 491–506. Zbl0534.60065MR736407
  7. [7] P. Révész. Random Walk in Random and Non-Random Environments, 2nd edition. World Scientific, Hackensack, NJ, 2005. Zbl1090.60001MR2168855
  8. [8] Z. Shi. A local time curiosity in random environment. Stochastic Process. Appl. 76 (1998) 231–250. Zbl0932.60054MR1642673
  9. [9] Z. Shi. Sinai’s walk via stochastic calculus. In Milieu aléatoires 53–74. F. Comets and E. Pardoux (Eds). Panoramas et Synthèses 12. Soc. Math. France, Paris, 2001. Zbl1031.60088MR2226845
  10. [10] Y. G. Sinai. The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27 (1982) 256–268. Zbl0505.60086MR657919
  11. [11] F. Solomon. Random walks in random environment. Ann. Probab. 3 (1975) 1–31. Zbl0305.60029MR362503
  12. [12] O. Zeitouni. Random walks in random environment. In XXXI Summer School in Probability, St Flour (2001) 193–312. Lecture Notes in Math. 1837. Springer, Berlin, 2004. Zbl1060.60103MR2071631

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