Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Metastable states in Brownian energy landscape

Dimitris Cheliotis

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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.


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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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