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.

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

## 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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## Additional indexing

Other titles: | |
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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; 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 15:48 |

Last Modified: | 05 Apr 2016 13:24 |

Publisher: | Institute of Mathematical Statistics |

ISSN: | 0091-1798 |

Publisher DOI: | https://doi.org/10.1214/009117904000000991 |

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