MARKOV-CHAINS WITH EXPONENTIALLY SMALL TRANSITION-PROBABILITIES - FIRST EXIT PROBLEM FROM A GENERAL DOMAIN .1. THE REVERSIBLE CASE

We consider general ergodic aperiodic Markov chains with finite state space whose transition probabilities between pairs of different commun icating states are exponentially small in a large parameter beta. We e xtend previous results by M. I. Freidlin and A. D. Wentzell (FW) on th e first exit problem from a general domain Q. In the present paper we analyze the case of reversible Markov chains. The general case will be studied in a forthcoming paper. We prove, in a purely probabilistic w ay and without using the FW graphical technique, some results on the f irst exit problem from a general domain Q containing many attractors. In particular we analyze the properties of special domains called cycl es and, by using the new concept of temporal entropy, we obtain new re sults leading to a complete description of the typical tube of traject ories during the first excursion outside Q.