Metastability: A potential theoretic approach

International Congress of Mathematicians, ICM 2006

Volumn 3, Issue , 2006, Pages 499-518

  Bovier, Anton ^a,b  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^b TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

Capacity; Markov processes; Metastability; Potential theory; Spectral theory

Indexed keywords

CAPACITY; DYNAMICAL BEHAVIOURS; GENERAL THEORY; METASTABILITIES; POTENTIAL THEORY; REVERSIBLE MARKOV PROCESS; SPECTRAL THEORY;

MARKOV PROCESSES;

EID: 84878042443     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (35)

1. Alonso, L., and Cerf, R., The three-dimensional polyominoes of minimal area. Electron. J. Combin. 3 (1996), Research Paper 27 (electronic).
2. Bovier, A., Metastability and ageing in stochastic dynamics. In Dynamics and randomness II (A. Maas, S. Martínez, J. San Martin, eds.), Nonlinear Phenomena and Complex Systems 10, Kluwer Academic Publishers, Dordrecht 2004, 17-80.
3. Bovier, A., Eckhoff, M., Gayrard, V., and Klein, M., Metastability in stochastic dynamics of disordered mean-field models. Probab. Theory Related Fields 119 (2001), 99-161.
4. Bovier, A., Manzo, F., Metastability in Glauber dynamics in the low-temperature limit: beyond exponential asymptotics. J. Statist. Phys. 107 (2002), 757-779.
5. Bovier, A., Eckhoff, M., Gayrard, V., and Klein, M., Metastability and low lying spectra in reversible Markov chains. Comm. Math. Phys. 228 (2002), 219-255.
6. Bovier, A., Eckhoff, M., Gayrard, V., and Klein, M., Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. (JEMS) 6 (2004), 399-424.
7. Bovier, A., Gayrard, V., and Klein, M., Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues. J. Eur. Math. Soc. (JEMS) 7 (2005), 69-99.
8. Bovier, A., den Hollander, F., and Nardi, F. R., Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary conditions. Probab. Theor. Rel. Fields. 135 (2) (2006), 265-310 .
9. BenArous, G., and Cerf, R., Metastability of the three-dimensional Ising model on a torus at very low temperatures. Electron. J. Probab. 1 (1996), 55 pp. (electronic).
10. Catoni, O., and Cerf, R., The exit path of a Markov chain with rare transitions. ESAIM Probab. Statist. 1 (1995/97), 95-144 (electronic).
11. Davies, E. B., Metastable states of symmetric Markov semigroups. I. Proc. Lond. Math. Soc. (3) 45 (1982), 133-150 .
12. Davies, E. B., Metastable states of symmetric Markov semigroups. II. J. Lond. Math. Soc. (2) 26 (1982), 541-556.
13. Davies, E. B., Spectral properties of metastable Markov semigroups. J. Funct. Anal. 52 (1983), 315-329.
14. Doyle, P. G., and Snell, J. L., Random walks and electrical networks. Carus Mathematical Monographs 22, Mathematical Association of America, Washington, DC, 1984.
15. Eyring, H., The activated complex in chemical reactions. J. Chem. Phys. 3 (1935), 107-115.
16. Freidlin, M. I., andWentzell, A. D., Random perturbations of dynamical systems. Second edition, Grundlehren Math.Wiss. 260, Springer-Verlag, NewYork 1998.
17. Gaveau, B., and Schulman, L. S., Theory of nonequilibrium first-order phase transitions for stochastic dynamics. J. Math. Phys. 39 (1998), 1517-1533.
18. Helffer, B., Klein, M., and Nier, F., Quantitative analysis of metastability in reversible diffusion processes via aWitten complex approach. Mat. Contemp. 26 (2004), 41-85.
19. Helffer, B., and Nier, F., Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Lecture Notes in Math. 1862, Springer-Verlag, Berlin 2005.
20. Helffer, B., and Nier, F., Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. The case with boundary. Preprint 04-40 IRMAR, Université de Rennes I, 2004.
21. den Hollander, F., Olivieri, E., and Scoppola, E., Metastability and nucleation for conservative dynamics. Probabilistic techniques in equilibrium and non-equilibrium statistical physics. J. Math. Phys. 41 (2000), 1424-1498.
22. den Hollander, F., Nardi, F. R., Olivieri, E., and Scoppola, E., Droplet growth for threedimensional Kawasaki dynamics. Probab. Theory Related Fields 125 (2003), 153-194.
23. den Hollander, F., Metastability under stochastic dynamics. Stochastic Process. Appl. 114 (2004), 1-26.
24. Holley, R. A., Kusuoka, S., and Stroock, W.S., Asymptotics of the spectral gap with applications to the theory of simulated annealing. J. Funct. Anal. 83 (1989), 333-347.
25. Huisinga, W., Meyn, S., and Schütte, Ch., Phase transitions and metastability for Markovian and molecular systems. Ann. Appl. Probab. 14 (2004), 419-458.
26. Kramers, H. A., Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7 (1949), 284-304.
27. Maier, R. S. and Stein, D. L., Limiting exit location distributions in the stochastic exit problem. SIAM J. Appl. Math. 57 (1997), 752-79.
28. Martinelli, F., On the kinetic Ising model below the critical temperature. In XIIIth International Congress on Mathematical Physics (London, 2000), International Press, Boston, MA, 2001, 297-301.
29. Martinelli, F., Relaxation times of Markov chains in statistical mechanics and combinatorial structures. In Probability on discrete structures, Encyclopaedia Math. Sci. 110, Springer- Verlag, Berlin 2004, 175-262.
30. Mathieu, P., Spectra, exit times, and long times asymptotics in the zero white noise limit. Stoch. Stoch. Rep. 55 (1995), 1-20.
31. Miclo, L., Comportement de spectres d'opérateurs de Schrödinger à basse température. Bull. Sci. Math. 119 (1995), 529-553.
32. Neves, E. J., and Schonmann, R. H., Critical droplets and metastability for a Glauber dynamics at very low temperature. Commun. Math. Phys. 137 (1991), 209-230.
33. Olivieri, E., and Scoppola, E., Markov chains with exponentially small transition probabilities: first exit problem from a general domain. I. The reversible case. J. Statist. Phys. 79 (1995), 613-647.
34. Olivieri, E., and Scoppola, E., Markov chains with exponentially small transition probabilities: first exit problem from a general domain. II. The general case. J. Statist. Phys. 84 (1996), 987-1041.
35. Olivieri, E., and Vares, M. E., Large deviations and metastability. Encyclopedia of Mathematics and its Applications 100, Cambridge University Press, Cambridge 2005.