Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Metastable states in Brownian energy landscape

Dimitris Cheliotis

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Abstract

Random walks and diffusions in symmetric random environment are known to exhibit metastable behavior: they tend to stay for long times in wells of the environment. For the case that the environment is a one-dimensional two-sided standard Brownian motion, we study the process of depths of the consecutive wells of increasing depth that the motion visits. When these depths are looked in logarithmic scale, they form a stationary renewal cluster process. We give a description of the structure of this process and derive from it the almost sure limit behavior and the fluctuations of the empirical density of the process.

Résumé

Il est bien connu que les marches aléatoires et les diffusions dans un environnement symétrique aléatoire ont un comportement métastable : elles tendent à rester longtemps dans les puits de l’environnement. Dans le cas où l’environnement est un mouvement brownien linéaire, nous étudions le processus des profondeurs des puits consécutifs de profondeur croissante que la dynamique visite. Quand ces profondeurs sont regardées à l’échelle logarithmique, elles forment un processus stationnaire de renouvellement. Nous donnons une description de la structure de ce processus et nous en déduisons le comportement asymptotique presque sûr et les fluctuations de sa densité empirique.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 3 (2015), 917-934.

Dates
Received: 20 August 2013
Revised: 24 February 2014
Accepted: 20 March 2014
First available in Project Euclid: 1 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1435759234

Digital Object Identifier
doi:10.1214/14-AIHP616

Mathematical Reviews number (MathSciNet)
MR3365967

Zentralblatt MATH identifier
1323.60133

Subjects
Primary: 60K37: Processes in random environments 60G55: Point processes 60F05: Central limit and other weak theorems

Keywords
Diffusion in random environment Brownian motion Excursion theory Renewal cluster process Confluent hypergeometric equation

Citation

Cheliotis, Dimitris. Metastable states in Brownian energy landscape. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 3, 917--934. doi:10.1214/14-AIHP616. https://projecteuclid.org/euclid.aihp/1435759234


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References

  • [1] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996.
  • [2] A. Bovier. Metastability: A potential theoretic approach. In Proceedings of the ICM 499–518. European Mathematical Society, Madrid, 2006.
  • [3] D. Cheliotis. Difusion in random environment and the renewal theorem. Ann. Probab. 33 (5) (2005) 1760–1781.
  • [4] D. Cheliotis and B. Virag. Patterns in Sinai’s walk. Ann. Probab. 41 (3B) (2013) 1900–1937.
  • [5] D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, 2nd edition. Springer, New York, 2003.
  • [6] P. Le Doussal, C. Monthus and D. Fisher. Random walkers in one-dimensional random environments: Exact renormalization group analysis. Phys. Rev. E (3) 59 (5) (1999) 4795–4840.
  • [7] R. Durrett. Probability: Theory and Examples, 4th edition. Cambridge Univ. Press, Cambridge, 2010.
  • [8] N. N. Lebedev. Special Functions and Their Applications. Prentice Hall, Engelwood Cliffs, NJ, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman. Unabridged and corrected republication.
  • [9] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Berlin, 1999.
  • [10] Z. Shi. Sinai’s walk via stochastic calculus. In Milieux Aléatoires 53–74. F. Comets and E. Pardoux (Eds). Panoramas et Synthèses 12. Société Mathématique de France, Paris, 2001.
  • [11] H. Tanaka. Limit theorem for one-dimensional diffusion process in Brownian environment. In Stochastic Analysis 156–172. Springer, Berlin, 1988.
  • [12] O. Zeitouni. Random Walks in Random Environment. In Lectures on Probability Theory and Statistics. Ecole d’Eté de Probabilités de Saint-Flour XXXI-2001 189–312. Lecture Notes in Math. 1837. Springer, Berlin, 2004.