 02371 A. Bovier, V. Gayrard, M. Klein
 Metastability in reversible diffusion processes II. Precise asymptotics for small eigenvalues
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Sep 11, 02

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Abstract. We continue the analysis of the
problem of metastability for reversible diffusion processes,
initiated in \cite{BEGK3}, with a precise analysis of the lowlying
spectrum of the generator. Recall that we are considering
processes with generators
of the form $\e \Delta +\nabla F(\cdot)\nabla$ 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 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 raduis $\e$ 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
\cite{BEGK3} it was proven
that these mean times are given, again up to mupltiplicative
errors that tend to one, by the classical {\it EyringKramers formula}.
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