# Convergence abrupte et métastabilité

Abstract : The aim of this thesis is to link two phenomena concerning the asymptotical behavior of stochastic processes, which were disjoined up to now. The abrupt convergence or cutoff phenomenon on one hand, and metastability on the other hand.
In the cutoff case an abrupt convergence towards the equilibrium measure occurs at a time which can be determined, whereas metastability is linked to a great uncertainty of the time at which we leave some equilibrium. We propose a common framework to compare and study both phenomena : that of discrete time birth and death chains on $\mathbb{N}$ with drift towards zero.
Under the drift hypothesis, we prove that there is an abrupt convergence towards zero, metastability in the other direction, and that the last exit in the metastability is the time reverse of a typical cutoff path.
We extend our approach to the Ehrenfest model, which allows us to prove abrupt convergence and metastability with a weaker drift hypothesis.
Keywords :
Document type :
Theses
Mathématiques [math]. Université de Rouen, 2007. Français
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https://tel.archives-ouvertes.fr/tel-00218132
Contributor : Olivier Bertoncini <>
Submitted on : Friday, January 25, 2008 - 1:56:21 PM
Last modification on : Thursday, January 11, 2018 - 6:12:27 AM
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• HAL Id : tel-00218132, version 1

### Citation

Olivier Bertoncini. Convergence abrupte et métastabilité. Mathématiques [math]. Université de Rouen, 2007. Français. 〈tel-00218132〉

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