94-230 E. Olivieri, E. Scoppola

MARKOV CHAINS WITH EXPONENTIALLY SMALL TRANSITION PROBABILITIES: FIRST EXIT PROBLEM FROM A GENERAL DOMAIN I. THE REVERSIBLE CASE. (90K, TeX) Jul 14, 94

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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

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