JournalsjemsVol. 6, No. 4pp. 399–424

Metastability in Reversible Diffusion Processes I: Sharp Asymptotics for Capacities and Exit Times

  • Véronique Gayrard

    CNRS Luminy, Marseille, France
  • Anton Bovier

    Weierstrass Institut für Angewandte Analysis und Stochastik, Berlin, Germany
  • Michael Eckhoff

    Universität Zürich, Switzerland
  • Markus Klein

    Universität Potsdam, Germany
Metastability in Reversible Diffusion Processes I: Sharp Asymptotics for Capacities and Exit Times cover
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Abstract

We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form \eΔ+F()-\e \Delta +\nabla F(\cdot)\nabla on Rd\R^d or subsets of Rd\R^d, where FF is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that {\it metastable exit times} from the attractive domains of the minima of FF can be related, up to multiplicative errors that tend to one as \e0\e\downarrow 0, to the capacities of suitably constructed sets. We show that this capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of FF at the starting minimum and the relevant {\it saddle points}. As a result, we are able to give the first rigorous proof of the classical {\it Eyring-Kramers formula} in dimension larger than 11. 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.

Cite this article

Véronique Gayrard, Anton Bovier, Michael Eckhoff, Markus Klein, Metastability in Reversible Diffusion Processes I: Sharp Asymptotics for Capacities and Exit Times. J. Eur. Math. Soc. 6 (2004), no. 4, pp. 399–424

DOI 10.4171/JEMS/14