In this paper we consider aperiodic ergodic Markov chains with transition probabilities exponentially small in a large parameter β. We extend to the general, not necessarily reversible case the analysis, started in part I of this work, of the first exit problem from a general domainQ containing many stable equilibria (attracting equilibrium points for the β=∞ dynamics). In particular we describe the tube of typical trajectories during the first excursion outsideQ.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
D. Chen, J. Feng, and M. Qian, The metastability of exponentally perturbed Markov chains,Chinese Sci. A 25(6):590–595 (1995).
T. S. Chiang and Y. Chow, A limit theorem for a class of inhomogeneous Markov processes,Ann. Prob. 17:1483–1502 (1989).
T. S. Chiang and Y. Chow, On the exit problem from a cycle of simulated annealing processes with application—A backward equation approach, Tech. Rept. Inst. Math. Academia Sinica (1995).
T. S. Chiang and Y. Chow, Asymptotic behavior of eigenvalues and random updating schemes,Appl. Math. Optim. 28:259–275 (1993).
M. I. Freidlin and A. D. Wentzell,Random Perturbations of Dynamical Systems, (Springer-Verlag, Berlin, 1984).
C. R. Hwang and S. J. Sheu, Singular perturbed Markov chains and exact behaviors of simulated annealing process,J. Theor. Prob. 5:223–249 (1992).
R. Kotecky and E. Olivieri, Droplet dynamics for asymmetric Ising model, preprint (1992).
R. Kotecky and E. Olivieri, Shapes of growing droplets—A model of escape from a metastable phase, in preparation.
E. J. Neves and R. H. Schonmann, Behaviour of droplets for a class of Glauber dynamics at very low temperatures,Commun. Math. Phys. 137:209 (1991).
E. J. Neves and R. H. Schonmann, Critical droplets and metastability for a Glauber dynamics at very low temperatures,Prob. Theory Related Fields 91:331 (1992).
E. Olivieri and E. Scoppola, Markov chains with exponentially small transition probabilities: First exit problem from a general domain—I. The reversible case, preprint.
E. Scoppola, Renormalization group for Markov chains and application to metastability,J. Stat. Phys. 73:83 (1993).
E. Scoppola, Renormalization and graph methods for Markov chains, inProceedings of the Conference “Advances in Dynamical Systems and Quantum Physics”—Capri 1993, in press.
A. Touvé, Cycle decompositions and simulated annealing, preprint LMENS-94.
A. Touvé, Rough large deviation estimates for the optimal convergence speed exponent of generalized simulated annealing algorithms,Ann. Inst. H. Poincaré (1995).
O. Catoni, Rough large deviation estimates for simulated annealing. Application to exponential schedules,Ann. Prob. 20:1109–1146 (1992).
R. Z. Has'minskii,Stochastic Stability of Differential Equations (Sijthoff and Noordhoff, 1980).
About this article
Cite this article
Olivieri, E., Scoppola, E. Markov chains with exponentially small transition probabilities: First exit problem from a general domain. II. The general case. J Stat Phys 84, 987–1041 (1996). https://doi.org/10.1007/BF02174126
- Markov chains
- first exit problem
- large deviations