Enzo Olivieri, Elisabetta Scoppola MARKOV CHAINS WITH EXPONENTIALLY SMALL TRANSITION PROBABILITIES: FIRST EXIT PROBLEM FROM A GENERAL DOMAIN (133K, plain tex) ABSTRACT. In this paper we consider aperiodic ergodic Markov chains with transition probabilities exponentially small in a large parameter $\b$ . We extend to the general not necessarily reversible case the analysis started in [OS] of the first exit problem from a general domain $Q$ containing many stable equilibria (attracting equilibrium points for the $\b = \infty$ dynamics). In particular we describe the tube of typical trajectories during the first excursion outside $Q$.