A. Bovier, M. Eckhoff, V. Gayrard, M. Klein
Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times.
(339K, PS)

ABSTRACT.  We develop a potential theoretic approach to the 
problem of metastability for reversible diffusion 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.
In analogy to previous work in discrete Markov chains, we show that 
{\it metastable exit times} from the attractive domains of the minima
of $F$  can be related,  up to mupltiplicative 
errors that tend to one as $\e\downarrow 0$, to the  capacities of
suitably constructed sets. We show that this capacities can be
computed, again   up to mupltiplicative 
errors that tend to one, in terms of local characteristics of $F$ 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 $1$. 
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.