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 communicating states are exponentially small in a large parameter beta. We extend previous results by M. I. Freidlin and A. D. Wentzell (FW) on the 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 way and without using the FW 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 cycles and, by using the new concept of temporal entropy, we obtain new results leading to a complete description of the typical tube of trajectories during the first excursion outside Q.