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

Anton Bovier;  Véronique Gayrard;  Markus Klein — 2005

Journal of the European Mathematical Society

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 − ϵ Δ + ∇ F ( · ) ∇ on ℝ d or subsets of 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...

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

Anton Bovier;  Michael Eckhoff;  Véronique Gayrard;  Markus Klein — 2004

Journal of the European Mathematical Society

We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form − ϵ Δ + ∇ F ( · ) ∇ 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 ϵ ↓ 0 , to the capacities of suitably constructed sets. We show that these capacities...

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