Electronic Journal of Probability

Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond

Nils Berglund and Barbara Gentz

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<p>We prove a Kramers-type law for metastable transition times for a class of one-dimensional parabolic stochastic partial differential equations (SPDEs) with bistable potential. The expected transition time between local minima of the potential energy depends exponentially on the energy barrier to overcome, with an explicit prefactor related to functional determinants. Our results cover situations where the functional determinants vanish owing to a bifurcation, thereby rigorously proving the results of formal computations announced in a previous work. The proofs rely on a spectral Galerkin approximation of the SPDE by a finite-dimensional system, and on a potential-theoretic approach to the computation of transition times in finite dimension.</p>

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 24, 58 pp.

Accepted: 16 February 2013
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35K57: Reaction-diffusion equations 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 37H20: Bifurcation theory [See also 37Gxx]

SPDEs reaction-diffusion equations metastability Kramers' law exit problem transition time large deviations Wentzell-Freidlin theory potential theory capacities Galerkin approximation subexponential asymptotics pitchfork bifurcation

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Berglund, Nils; Gentz, Barbara. Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond. Electron. J. Probab. 18 (2013), paper no. 24, 58 pp. doi:10.1214/EJP.v18-1802. https://projecteuclid.org/euclid.ejp/1465064249

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