Permanent URL to this publication: http://dx.doi.org/10.5167/uzh-21708

**Eckhoff, M (2005). Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime. The Annals of Probability, 33(1):244-299.**

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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 |

DDC: | 510 Mathematics |

Uncontrolled Keywords: | Capacity; eigenvalue problem; exit problem; exponential distribution; diffusion process; ground-state 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 16:48 |

Last Modified: | 27 Nov 2013 22:35 |

Publisher: | Institute of Mathematical Statistics |

ISSN: | 0091-1798 |

Publisher DOI: | 10.1214/009117904000000991 |

Citations: | Web of Science® Google Scholar™ |

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