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Valleys and the Maximum Local Time for Random Walk in Random Environment
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  • Published: 27 April 2006

Valleys and the Maximum Local Time for Random Walk in Random Environment

  • Amir Dembo1,
  • Nina Gantert2,
  • Yuval Peres3 &
  • …
  • Zhan Shi4 

Probability Theory and Related Fields volume 137, pages 443–473 (2007)Cite this article

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  • 13 Citations

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Abstract

Let ξ (n, x) be the local time at x for a recurrent one-dimensional random walk in random environment after n steps, and consider the maximum ξ*(n) = max x ξ(n, x). It is known that lim sup \(_n\xi^*(n)/n\) is a positive constant a.s. We prove that lim inf \(_n(log\!!!log\!!!log\!!!n) \xi^*(n)/n\) is a positive constant a.s. this answers a question of P. Révész [5]. The proof is based on an analysis of the valleys in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time n large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.

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References

  1. Borodin A.N., Salminen P. (2002) Handbook of Brownian motion – facts and formulae, 2nd edn. Birkhäuser, Basel

    MATH  Google Scholar 

  2. Doyle, P.G., Snell, E.J. Probability: random walks and electrical networks. Carus Math Monographs 22, Math Assoc Amer, Washington DC 1984

  3. Gantert N., Shi Z. (2002) Many visits to a single site by a transient random walk in random environment. Stoch Proc Appl 99, 159–176

    Article  MathSciNet  Google Scholar 

  4. Golosov A.O. (1984) Localization of random walks in one-dimensional random environments. Comm Math Phys 92, 491–506

    Article  MathSciNet  Google Scholar 

  5. Révész P. (1990) Random walk in random and non-random environments. World Scientific, Singapore

    MATH  Google Scholar 

  6. Revuz D., Yor M. (1999) Continuous martingales and Brownian motion: 3rd edn. Grundlehren der Mathematischen Wissenschaften, vol. 293. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  7. Shi Z. (1998) A local time curiosity in random environment. Stoch Proc Appl 76, 231–250

    Article  Google Scholar 

  8. Shiryaev A.N. (1996) Probability 2nd edn. Springer, Berlin Heidelberg New York

    Google Scholar 

  9. Sinai Ya.G. (1982) The limiting behavior of a one-dimensional random walk in a random medium. Theor Probab Appl 27, 256–268

    Article  MathSciNet  Google Scholar 

  10. Zeitouni O. (2004) Random walks in random environment, XXXI summer school in probability, St Flour (2001), Lecture Notes in Mathematics 1837, vol. 1837. pp. 193–312. Springer, Berlin Heidelberd New York

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Amir Dembo

  2. Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstrasse 62, D-48149, Müenster, Germany

    Nina Gantert

  3. Department of Statistics, University of California Berkeley, 367 Evans Hall, Berkeley, C.A, 94720-3860, USA

    Yuval Peres

  4. Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, 4 place Jussieu, F-75252, Paris Cedex 05, France

    Zhan Shi

Authors
  1. Amir Dembo
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  2. Nina Gantert
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  3. Yuval Peres
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  4. Zhan Shi
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Corresponding author

Correspondence to Amir Dembo.

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Dembo, A., Gantert, N., Peres, Y. et al. Valleys and the Maximum Local Time for Random Walk in Random Environment. Probab. Theory Relat. Fields 137, 443–473 (2007). https://doi.org/10.1007/s00440-006-0005-6

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  • Received: 12 September 2005

  • Revised: 07 March 2006

  • Published: 27 April 2006

  • Issue Date: March 2007

  • DOI: https://doi.org/10.1007/s00440-006-0005-6

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Mathematics Subject Classification (2000)

  • 60K37
  • 60G50
  • 60J55
  • 60F10

Keywords

  • Random walk in random environment
  • Local time
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