Permanent URL to this publication: http://dx.doi.org/10.5167/uzh21708
Eckhoff, M (2005). Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime. The Annals of Probability, 33(1):244299.

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Abstract
We investigate the close connection between metastability of the reversible diffusion process X defined by the stochastic differential equation
d Xt = −∇ F (Xt ) d t + √2ε d Wt , ε > 0,
and the spectrum near zero of its generator −Lɛ≡ɛΔ−∇F⋅∇, where F:ℝd→ℝ and W denotes Brownian motion on ℝd. For generic F to each local minimum of F there corresponds a metastable state. We prove that the distribution of its rescaled relaxation time converges to the exponential distribution as ɛ↓0 with optimal and uniform error estimates. Each metastable state can be viewed as an eigenstate of Lɛ with eigenvalue which converges to zero exponentially fast in 1/ɛ. Modulo errors of exponentially small order in 1/ɛ this eigenvalue is given as the inverse of the expected metastable relaxation time. The eigenstate is highly concentrated in the basin of attraction of the corresponding trap.
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Item Type:  Journal Article, refereed, original work 
Communities & Collections:  07 Faculty of Science > Institute of Mathematics 
Dewey Decimal Classification:  510 Mathematics 
Uncontrolled Keywords:  Capacity; eigenvalue problem; exit problem; exponential distribution; diffusion process; groundstate splitting; large deviations; metastability; relaxation time; reversibility; potential theory; Perron–Frobenius eigenvalues; semiclassical limit; Witten’s Laplace 
Language:  English 
Date:  2005 
Deposited On:  19 Feb 2010 15:48 
Last Modified:  27 Nov 2013 21:35 
Publisher:  Institute of Mathematical Statistics 
ISSN:  00911798 
Publisher DOI:  10.1214/009117904000000991 
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