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 β. We extend previous results by M. I. Freidlin and A. D. Wentzell (FW) on the first exit problem from a general domainQ. In the present paper we analyze the case ofreversible Markov chains. The general case will be studied in a forthcoming paper. We probe, in a purely probabilistic way and without using the FW graphical technique, some results on the first exit problem from a general domainQ containing many attractors. In particular we analyze the properties of special domains calledcycles and, by using the new concept oftemporal entropy, we obtain new results leading to a complete description of the typical tube of trajectories during the first excursion outsideQ.
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Dedicated to the memory of Claude Kipnis.
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Olivieri, E., Scoppola, E. Markov chains with exponentially small transition probabilities: First exit problem from a general domain. I. The reversible case. J Stat Phys 79, 613–647 (1995). https://doi.org/10.1007/BF02184873
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DOI: https://doi.org/10.1007/BF02184873