# Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times

Anton Bovier; Michael Eckhoff; Véronique Gayrard; Markus Klein

Journal of the European Mathematical Society (2004)

- Volume: 006, Issue: 4, page 399-424
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topBovier, Anton, et al. "Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times." Journal of the European Mathematical Society 006.4 (2004): 399-424. <http://eudml.org/doc/277604>.

@article{Bovier2004,

abstract = {We develop a potential theoretic approach to the problem of metastability for reversible diffusion 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. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of $F$ can be
related, up to multiplicative errors that tend to one as $\epsilon \downarrow 0$, to the capacities of suitably constructed
sets. We show that these capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of $F$ at the starting minimum and the relevant saddle points. As a result, we are able to give the first rigorous proof of the classical Eyring–Kramers formula in dimension larger than 1. The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes.},

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

journal = {Journal of the European Mathematical Society},

keywords = {metastability; reversible diffusion processes; potential theory; capacity; exit times; Ito stochastic differential equation; reversible diffusion processes; Ito stochastic differential equation},

language = {eng},

number = {4},

pages = {399-424},

publisher = {European Mathematical Society Publishing House},

title = {Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times},

url = {http://eudml.org/doc/277604},

volume = {006},

year = {2004},

}

TY - JOUR

AU - Bovier, Anton

AU - Eckhoff, Michael

AU - Gayrard, Véronique

AU - Klein, Markus

TI - Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times

JO - Journal of the European Mathematical Society

PY - 2004

PB - European Mathematical Society Publishing House

VL - 006

IS - 4

SP - 399

EP - 424

AB - We develop a potential theoretic approach to the problem of metastability for reversible diffusion 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. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of $F$ can be
related, up to multiplicative errors that tend to one as $\epsilon \downarrow 0$, to the capacities of suitably constructed
sets. We show that these capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of $F$ at the starting minimum and the relevant saddle points. As a result, we are able to give the first rigorous proof of the classical Eyring–Kramers formula in dimension larger than 1. The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes.

LA - eng

KW - metastability; reversible diffusion processes; potential theory; capacity; exit times; Ito stochastic differential equation; reversible diffusion processes; Ito stochastic differential equation

UR - http://eudml.org/doc/277604

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.