Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times

Journal of the European Mathematical Society

Volumn 6, Issue 4, 2004, Pages 399-424

  Bovier, Anton ^a,b   Eckhoff, Michael ^c   Gayrard, Véronique ^d,e,f   Klein, Markus ^g  

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

^b TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

^c UNIVERSITY OF ZURICH   (Switzerland)

^d EPFL   (Switzerland)

^e UNIVERSITÉ DE MONTRÉAL   (Canada)

^f CNRS   (France)

^g UNIVERSITY OF POTSDAM   (Germany)

Author keywords

Capacity; Diffusion processes; Exit times; Metastability; Potential theory

Indexed keywords

EID: 4644347796     PISSN: 14359855     EISSN: None     Source Type: Journal    
DOI: 10.4171/JEMS/14     Document Type: Article

References (28)

1. [BluGet] Blumenthal, R. M., Getoor, R. K.: Markov Processes and Potential Theory. Academic Press, New York (1968) Zbl 0169.49204 MR 41 #9348
2. [BEGK1] Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in stochastic dynamics of disordered mean-field models. Probab. Theory Related Fields 119, 99-161 (2001) Zbl 1012.82015 MR 2001k:82096
3. [BEGK2] Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability and low-lying spectra in reversible Markov chains. Comm. Math. Phys. 228, 219-255 (2002) Zbl 1010.60088 MR 1911735
4. [BGK] Bovier, A., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes II. Precise asymptotics for small eigenvalues. J. Eur. Math. Soc. (to appear)
5. [BuMa1] Buslov, V. A., Makarov, K. A.: A time-scale hierarchy with small diffusion. Teoret. Mat. Fiz. 76, 219-230 (1988) (in Russian); English transl.: Theoret. Math. Phys. 76, 818-826 (1989) Zbl 0698.35076 MR 90a:58188
6. [BuMa1] Buslov, V. A., Makarov, K. A.: A time-scale hierarchy with small diffusion. Teoret. Mat. Fiz. 76, 219-230 (1988) (in Russian); English transl.: Theoret. Math. Phys. 76, 818-826 (1989) Zbl 0698.35076 MR 90a:58188
7. [BuMa2] Buslov, V. A., Makarov, K. A.: Life spans and least eigenvalues of an operator of small diffusion. Mat. Zametki 51, 20-31 (1992) (in Russian); English transl.: Math. Notes 51, 14-21 (1992) Zbl 0755.34086 MR93c:34161
8. [BuMa2] Buslov, V. A., Makarov, K. A.: Life spans and least eigenvalues of an operator of small diffusion. Mat. Zametki 51, 20-31 (1992) (in Russian); English transl.: Math. Notes 51, 14-21 (1992) Zbl 0755.34086 MR93c:34161
9. [Doo] Doob, J. L.: Classical Potential Theory and its Probabilistic Counterpart. Grundlehren Math. Wiss. 262, Springer, Berlin (1984) Zbl 0549. 31001 MR 85k:31001
10. [DS] Doyle, P. G., Snell, J. L., Random Walks and Electrical Networks. Carus Math. Monogr. 22, Math. Assoc. America, Washington, DC (1984) Zbl 0583.60065 MR 89a:9402
11. [Ey] Eyring, H.: The activated complex in chemical reactions. J. Chem. Phys. 3, 107-115 (1935)
12. [FW] Freidlin, M. I., Wentzell, A. D.: Random Perturbations of Dynamical Systems. Springer, Berlin (1984) Zbl 0922.60006 MR 99h:60128
13. [GT] Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. 2nd ed., Grundlehren Math. Wiss. 224, Springer, Berlin (1983) Zbl 0562.35001 MR 86c:35035
14. [GLE] Glasstone, S., Laidler, K. J., Eyring, H.: The Theory of Rate Processes. McGraw-Hill, New York (1941)
15. [HS1] Helffer, B., Sjöstrand, J.: J. Multiple wells in the semiclassical limit. I. Comm. Partial Differential Equations 9, 337-408 (1984) Zbl 0546.35053 MR 86c:35113
16. [HS2] Helffer, B., Sjöstrand, J.: Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation. Ann. Inst. H. Poincaré Phys. Théor. 42, 127-212 (1985) Zbl 0595.35031 MR87a:35142
17. [HS3] Helffer, B., Sjöstrand, J.: Multiple wells in the semiclassical limit. III. Interaction through nonresonant wells. Math. Nachr. 124, 263-313 (1985) Zbl 0597.35023 MR 87i:35161
18. [HS4] Helffer, B., Sjöstrand, J.: Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten. Comm. Partial Differential Equations 10, 245-340 (1985) Zbl 0597.35024 MR 87i:35162
19. [Kaku] Kakutani, S.: Markoff process and the Dirichlet problem. Proc. Japan Acad. 21, 227-233 (1945) MR 0032866
20. [KN] Kipnis, C., Newman, C. M., The metastable behavior of infrequently observed, weakly random, one-dimensional diffusion processes. SIAM J. Appl. Math. 45, 972-982 (1985) Zbl 0592.60063 MR 87j:60112
21. [Kolo] Kolokoltsov, V. N., Semiclassical Analysis for Diffusions and Stochastic Processes. Springer, Berlin (2000) Zbl 0951.60001 MR 2001f:58073
22. [KoMak] Kolokoltsov, V. N., Makarov, K. A., Asymptotic spectral analysis of a small diffusion operator and the life times of the corresponding diffusion process. Russian J. Math. Phys. 4, 341-360 (1996) Zbl 0912.58042 MR 99b:58228
23. [Kra] Kramers, H. A., Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284-304 (1940) Zbl 0061.46405 MR 0002962
24. [MS1] Maier, R. S., Stein, D. L., Limiting exit location distributions in the stochastic exit problem. SIAM J. Appl. Math. 57, 752-790 (1997) Zbl 0874.60072 MR98a:60112
25. [MatSch1] Matkowsky, B. J., Schuss, Z.: The exit problem: a new approach to diffusion across potential barriers. SIAM J. Appl. Math. 36, 604-623 (1979) Zbl 0406.60071 MR 80c:82011
26. [NS] Nash-Williams, C. St. J. A.: Random walk and electric currents in networks. Proc. Cambridge Philos. Soc. 55, 181-194 (1959) Zbl 0100.13602 MR 23 #A2239
27. [Szni] Sznitman, A.-S.: Brownian Motion, Obstacles and Random Media. Springer Monogr. Math., Springer, Berlin (1998) Zbl 0973.60003 MR 2001h:60147
28. [Wen] Wentzell, A. D.: Formulas for eigenfunctions and eigenmeasures that are connected with a Markov process. Teor. Veroyatnost. i Primenen. 18, 3-29 (1973) (in Russian) MR 0321200