Metastability in reversible diffusion processes. II: Precise asymptotics for small eigenvalues
type: Item
instance of: scholarly article
title: Metastability in reversible diffusion processes. II: Precise asymptotics for small eigenvalues
zbMATH Open document ID: 1105.82025
DOI: 10.4171/JEMS/22
author: Anton Bovier
author: V. Gayrard
author: M. Klein
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2005-03-21T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_abstract.php
review text: Summary: We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in Part I [ibid. 6, No. 4, 399--424 (2004; Zbl 1076.82045)], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form \(-\varepsilon\Delta+\nabla F(\cdot)\nabla\) on \(\mathbb R^d\) or subsets of \(\mathbb R^d\), where \(F\) is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of the spectrum is given, up to multiplicative errors tending to one, by the eigenvalues of the classical capacity matrix of the array of capacitors made of balls of radius \(\varepsilon\) centered at the positions of the local minima of \(F\). We also get very precise uniform control on the corresponding eigenfunctions. Moreover, these eigenvalues can be identified with the same precision with the inverse mean metastable exit times from each minimum. In Part I it was proven that these mean times are given, again up to multiplicative errors that tend to one, by the classical Eyring-Kramers formula.
Mathematics Subject Classification ID: 82C44
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 2148277
zbMATH Keywords: metastability
zbMATH Keywords: diffusion processes
zbMATH Keywords: spectral theory
zbMATH Keywords: potential theory
zbMATH Keywords: capacity
zbMATH Keywords: exit times
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2148277