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.
@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] Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl
.[2] Metastability: A potential theoretic approach. In Proceedings of the ICM 499–518. European Mathematical Society, Madrid, 2006. | MR | Zbl
.[3] Difusion in random environment and the renewal theorem. Ann. Probab. 33 (5) (2005) 1760–1781. | MR | Zbl
.[4] Patterns in Sinai’s walk. Ann. Probab. 41 (3B) (2013) 1900–1937. | MR | Zbl
and .[5] An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, 2nd edition. Springer, New York, 2003. | MR | Zbl
and .[6] Random walkers in one-dimensional random environments: Exact renormalization group analysis. Phys. Rev. E (3) 59 (5) (1999) 4795–4840. | MR
, and .[7] Probability: Theory and Examples, 4th edition. Cambridge Univ. Press, Cambridge, 2010. | DOI | MR | Zbl
.[8] 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] Continuous Martingales and Brownian Motion, 3rd edition. Springer, Berlin, 1999. | DOI | MR | Zbl
and .[10] 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] Limit theorem for one-dimensional diffusion process in Brownian environment. In Stochastic Analysis 156–172. Springer, Berlin, 1988. | MR | Zbl
.[12] 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 :