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

Bovier, 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 -