# Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues

Anton Bovier; Véronique Gayrard; Markus Klein

Journal of the European Mathematical Society (2005)

- Volume: 007, Issue: 1, page 69-99
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topBovier, Anton, Gayrard, Véronique, and Klein, Markus. "Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues." Journal of the European Mathematical Society 007.1 (2005): 69-99. <http://eudml.org/doc/277203>.

@article{Bovier2005,

abstract = {We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form $−\epsilon \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 $\epsilon $ 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 [BEGK3] it was proven that these mean times are given, again up to multiplicative errors that tend to one, by the classical
Eyring–Kramers formula.},

author = {Bovier, Anton, Gayrard, Véronique, Klein, Markus},

journal = {Journal of the European Mathematical Society},

keywords = {metastability; diffusion processes; spectral theory; potential theory; capacity; exit times; metastability; diffusion processes; spectral theory; potential theory; capacity; exit times},

language = {eng},

number = {1},

pages = {69-99},

publisher = {European Mathematical Society Publishing House},

title = {Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues},

url = {http://eudml.org/doc/277203},

volume = {007},

year = {2005},

}

TY - JOUR

AU - Bovier, Anton

AU - Gayrard, Véronique

AU - Klein, Markus

TI - Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues

JO - Journal of the European Mathematical Society

PY - 2005

PB - European Mathematical Society Publishing House

VL - 007

IS - 1

SP - 69

EP - 99

AB - We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form $−\epsilon \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 $\epsilon $ 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 [BEGK3] it was proven that these mean times are given, again up to multiplicative errors that tend to one, by the classical
Eyring–Kramers formula.

LA - eng

KW - metastability; diffusion processes; spectral theory; potential theory; capacity; exit times; metastability; diffusion processes; spectral theory; potential theory; capacity; exit times

UR - http://eudml.org/doc/277203

ER -

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.