WIAS Preprint No. 768, (2002)

Metastability in reversible diffusion processes II. Precise asymptotics for small eigenvalues



Authors

  • Bovier, Anton
  • Gayrard, Veronique
  • Klein, Markus

2010 Mathematics Subject Classification

  • 82C44 60K35

Keywords

  • Metastability, diffusion processes, spectral theory, potential theory, capacity, exit times

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. 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 small balls 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.

Appeared in

  • J.Eur.Math.Soc. (JEMS) vol. 7, pp. 69--99, 2005

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