Metastable states in brownian energy landscape
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 917-934.

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.

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.

DOI : 10.1214/14-AIHP616
Mots clés : diffusion in random environment, brownian motion, excursion theory, renewal cluster process, confluent hypergeometric equation
@article{AIHPB_2015__51_3_917_0,
     author = {Cheliotis, Dimitris},
     title = {Metastable states in brownian energy landscape},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {917--934},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {3},
     year = {2015},
     doi = {10.1214/14-AIHP616},
     mrnumber = {3365967},
     zbl = {1323.60133},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/14-AIHP616/}
}
TY  - JOUR
AU  - Cheliotis, Dimitris
TI  - Metastable states in brownian energy landscape
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2015
SP  - 917
EP  - 934
VL  - 51
IS  - 3
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/14-AIHP616/
DO  - 10.1214/14-AIHP616
LA  - en
ID  - AIHPB_2015__51_3_917_0
ER  - 
%0 Journal Article
%A Cheliotis, Dimitris
%T Metastable states in brownian energy landscape
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2015
%P 917-934
%V 51
%N 3
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/14-AIHP616/
%R 10.1214/14-AIHP616
%G en
%F AIHPB_2015__51_3_917_0
Cheliotis, Dimitris. Metastable states in brownian energy landscape. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 917-934. doi : 10.1214/14-AIHP616. http://www.numdam.org/articles/10.1214/14-AIHP616/

[1] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl

[2] A. Bovier. Metastability: A potential theoretic approach. In Proceedings of the ICM 499–518. European Mathematical Society, Madrid, 2006. | MR | Zbl

[3] D. Cheliotis. Difusion in random environment and the renewal theorem. Ann. Probab. 33 (5) (2005) 1760–1781. | MR | Zbl

[4] D. Cheliotis and B. Virag. Patterns in Sinai’s walk. Ann. Probab. 41 (3B) (2013) 1900–1937. | MR | Zbl

[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. | MR | Zbl

[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. | MR

[7] R. Durrett. Probability: Theory and Examples, 4th edition. Cambridge Univ. Press, Cambridge, 2010. | DOI | MR | Zbl

[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. | MR | Zbl

[9] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Berlin, 1999. | DOI | MR | Zbl

[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. | MR | Zbl

[11] H. Tanaka. Limit theorem for one-dimensional diffusion process in Brownian environment. In Stochastic Analysis 156–172. Springer, Berlin, 1988. | MR | Zbl

[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. | MR | Zbl

Cité par Sources :