TY  - JOUR
A1  - Bovier, Anton
A1  - Gayrard, Veronique
A1  - Klein, Markus
T1  - Metastability in reversible diffusion processes : II. Precise asymptotics for small eigenvalues
N2  - 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 -epsilonDelta + delF(.) del on R-d or subsets of Rd, 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
Y1  - 2005
SN  - 1435-9855
ER  - 
TY  - JOUR
A1  - Bovier, Anton
A1  - Eckhoff, Michael
A1  - Gayrard, Veronique
A1  - Klein, Markus
T1  - Metastability in reversible diffusion processes : I. Sharp asymptotics for capacities and exit times
N2  - We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form -epsilonDelta+ delF(.) del on R-d or subsets of 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 down arrow 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
Y1  - 2004
SN  - 1435-9855
ER  - 
TY  - JOUR
A1  - Bovier, Anton
A1  - Eckhoff, Michael
A1  - Gayrard, Veronique
A1  - Klein, Markus
T1  - Metastability and low-Lying spectra in reversible Markov chains
Y1  - 2002
ER  - 
TY  - JOUR
A1  - Bovier, Anton
A1  - Eckhoff, Michael
A1  - Gayrard, Veronique
A1  - Klein, Markus
T1  - Metastability in stochastic dynamics of disordered mean-field models
Y1  - 2001
SN  - 0178-8051
ER  -