Phase transitions and metastability in Markovian and molecular systems

Annals of Applied Probability

Volumn 14, Issue 1, 2004, Pages 419-458

  Huisinga, Wilhelm ^a   Meyn, Sean ^b,c   Schütte, Christof ^a  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

^b UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^c University of Illinois at Urbana Champaign   (United States)

Author keywords

Large deviations; Markov process

Indexed keywords

EID: 12844274920     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/aoap/1075828057     Document Type: Article

References (42)

1. BALAJI, S. and MEYN, S. P. (2000). Multiplicative ergodicity and large deviations for an irreducible Markov chain. Stochastic Process. Appl. 90 123-144.
2. BOLTHAUSEN, E., DEUSCHEL, J.-D. and TAMURA, Y. (1995). Laplace approximations for large deviations of nonreversible Markov processes. The nondegenerate case. Ann. Probab. 23 236-267.
3. BOVIER, A., ECKHOFF, M., GAYRARD, V. and KLEIN, M. (2001). Metastability in stochastic dynamics of disordered mean-field models. Probab. Theory Related Fields 119 99-161.
4. BOVIER, A., ECKHOFF, M., GAYRARD, V. and KLEIN, M. (2002). Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. Technical report.
5. BOVIER, A., ECKHOFF, M., GAYRARD, V. and KLEIN, M. (2002). Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues. Technical report.
6. BOVIER, A. and MANZO, F. (2001). Metastability in Glauber dynamics in the low-temerature limit: Beyond exponential asymptotics. Preprint, Weierstrass-Institute für Angewandte Analysis und Stochastik.
7. DEUFLHARD, P., HUISINGA, W., FISCHER, A. and SCHÜTTE, CH. (2000). Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Linear Algebra Appl. 315 39-59.
8. DONSKER, M. D. and VARADHAN, S. R. S. (1983). Asymptotic evaluation of certain Markov process expectations for large time. IV. Comm. Pure Appl. Math. 36 183-212.
9. DOWN, D., MEYN, S. P. and TWEEDIE, R. L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 1671-1691.
10. FENG, J. and KURTZ, T. G. (2000). Large deviations for stochastic processes. Preprint.
11. FLEMING, W. H. (1978). Exit probabilities and optimal stochastic control. Appl. Math. Optim. 4 329-346.
12. FLEMING, W. H. and JAMES, M. R. (1992). Asymptotic series and exit time probabilities. Ann. Probab. 20 1369-1384.
13. FLEMING, W. H. and MCENEANEY, W. M. (1995). Risk-sensitive control on an infinite time horizon. SIAM J. Control Optim. 33 1881-1915.
14. FLEMING, W. H. and SHEU, S.-J. (1997). Asymptotics for the principal eigenvalue and eigenfunction of a nearly first-order operator with large potential. Ann. Probab. 25 1953-1994.
15. FREIDLIN, M. I. and WENTZELL, A. D. (1984). Random Perturbations of Dynamical Systems. Springer, New York.
16. GLYNN, P. W. and THORISSON, H. (2001). Two-sided taboo limits for Markov processes and associated perfect simulation. Stochastic Process. Appl. 91 1-20.
17. GLYNN, P. W. and THORISSON, H. (2002). Structural characterization of taboo-stationarity for general processes in two-sided time. Stochastic Process. Appl. 102 311-318.
18. HUISINGA, W. (2001). Metastability of Markovian systems: A transfer operator approach in application to molecular dynamics. Ph.D. thesis, Free Univ. Berlin.
19. JENSEN, J. L. (1987). A note on asymptotic expansions for Markov chains using operator theory. Adv. in Appl. Math. 8 377-392.
20. KONTOYIANNIS, I. and MEYN, S. P. (2002). Large deviation asymptotics and the spectral theory of multiplicatively regular Markov processes. Submitted for publication.
21. KONTOYIANNIS, I. and MEYN, S. P. (2003). Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13 304-362.
22. KUNITA, H. (1978). Supports of diffusion processes and controllability problems. In Proceedings of the International Symposium on Stochastic Differential Equations (K. Itô, ed.) 163-185. Wiley, New York.
23. KUNITA, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press.
24. LIBERZON, D. and BROCKETT, R. W. (2000). Spectral analysis of Fokker-Planck and related operators arising from linear stochastic differential equations. SIAM J. Control Optim. 38 1453-1467.
25. LOBRY, C. (1970). Contrôlabilité des systèmes non linéaires. SIAM J. Control 8 573-605.
26. MATTINGLY, J., STUART, A. M. and HIGHAM, D. J. (2001). Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerated noise. To appear.
27. MEYN, S. P. and TWEEDIE, R. L. (1993). Generalized resolvents and Harris recurrence of Markov processes. Contemp. Math. 149 227-250.
28. MEYN, S. P. and TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
29. MEYN, S. P. and TWEEDIE, R. L. (1993). Stability of Markovian processes II: Continuous time processes and sampled chains. Adv. in Appl. Probab. 25 487-517.
30. NEVEU, J. (1964). Chaînes de Markov et théorie du potentiel. Ann. Fac. Sci. Univ. Clermont-Ferrand 24 37-89.
31. NUMMELIN, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Univ. Press.
32. REY-BELLET, L. and THOMAS, L. E. (2000). Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators. Comm. Math. Phys. 215 1-24.
33. REY-BELLET, L. and THOMAS, L. E. (2001). Fluctuations of the entropy production in anharmonic chains. To appear.
34. SCHÜTTE, CH., FISCHER, A., HUISINGA, W. and DEUFLHARD, P. (1999). A direct approach to conformational dynamics based on hybrid Monte Carlo. J. Comput. Phys. Special Issue on Computational Biophysics 151 146-168.
35. SCHÜTTE, CH. and HUISINGA, W. (2000). On conformational dynamics induced by Langevin processes. In EQUADIFF 99-International Conference on Differential Equations (B. Fiedler, K. Gröger and J. Sprekels, eds.) 2 1247-1262. World Scientific, Singapore.
36. SCHÜTTE, CH., HUISINGA, W. and DEUFLHARD, P. (2001). Transfer operator approach to conformational dynamics in biomolecular systems. In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (B. Fiedler, ed.). Springer, New York.
37. SENETA, E. and VERE-JONES, D. (1966). On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probab. 3 403-434.
38. STROOCK, D. W. and VARADHAN, S. R. S. (1972). On the support of diffusion processes with applications to the strong maximum principle. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 333-359. Univ. California Press, Berkeley.
39. SUSSMANN, H. J. and JURDJEVIC, V. (1972). Controllability of nonlinear systems. J. Differential Equations 12 95-116.
40. THORISSON, H. (2000). Coupling, Stationarity and Regeneration. Springer, New York.
41. TWEEDIE, R. L. (1974). Quasi-stationary distributions for Markov chains on a general state space. J. Appl. Probab. 11 726-741.
42. VARADHAN, S. R. S. (1984). Large Deviations and Applications. SIAM, Philadelphia, PA.