E. Olivieri, E. Scoppola
 MARKOV CHAINS WITH EXPONENTIALLY SMALL TRANSITION PROBABILITIES:
FIRST EXIT PROBLEM FROM A GENERAL DOMAIN
I. THE REVERSIBLE CASE.
(90K, TeX)

ABSTRACT.  We consider  general ergodic aperiodic Markov chains with finite state space
whose transition probabilities between pairs of different communicating
states are exponentially small in a large parameter $\beta$.\par
We extend previous results by Freidlin and Wentzell ( [FW] ) on the first
exit problem from a general domain $Q$. \par
In the present paper we analyze the case of {\it reversible} Markov chains.
The general case will be studied in a forthcoming paper.\par
We prove, in a purely probabilistic way and without using F-W graphical
technique, some results on the first exit problem from a general domain $Q$
containing many attractors. In particular we
analyze the properties of special domains called {\it cycles } and,
by using the new concept of {\it temporal entropy}, we
obtain new results
leading to a complete description of
the typical tube of trajectories during the first excursion outside
$Q$.\par