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Metastability in reversible diffusion processes. II: Precise asymptotics for small eigenvalues. (English) Zbl 1105.82025

Summary: We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in Part I [ibid. 6, No. 4, 399–424 (2004; Zbl 1076.82045)], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form \(-\varepsilon\Delta+\nabla F(\cdot)\nabla\) on \(\mathbb R^d\) or subsets of \(\mathbb R^d\), where \(F\) is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of the spectrum is given, up to multiplicative errors tending to one, by the eigenvalues of the classical capacity matrix of the array of capacitors made of balls of radius \(\varepsilon\) centered at the positions of the local minima of \(F\). We also get very precise uniform control on the corresponding eigenfunctions. Moreover, these eigenvalues can be identified with the same precision with the inverse mean metastable exit times from each minimum. In Part I it was proven that these mean times are given, again up to multiplicative errors that tend to one, by the classical Eyring-Kramers formula.

MSC:

82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
60J45 Probabilistic potential theory
60J60 Diffusion processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory

Citations:

Zbl 1076.82045
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References:

[1] 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
[2] 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
[3] Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times, J. Eur. Math. Soc. 6 , 399- 424 (2004) · Zbl 1076.82045
[4] Blumenthal, R. M., Getoor, R. K.: Markov Processes and Potential Theory, Academic Press, New York (1968) · Zbl 0169.49204
[5] 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
[6] 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
[7] Davies, E. B.: Metastable states of symmetric Markov semigroups. I. Proc. London Math. Soc. 45 , 133-150 (1982) · Zbl 0498.47017
[8] Davies, E. B.: Metastable states of symmetric Markov semigroups. II, J. London Math. Soc. 26 , 541-556 (1982) · Zbl 0527.47028
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