UZH-Logo

Maintenance Infos

Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime


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.

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.

Altmetrics

Downloads

50 downloads since deposited on 19 Feb 2010
0 downloads since 12 months
Detailed statistics

Additional indexing

Other titles:
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:10.1214/009117904000000991
Permanent URL: http://doi.org/10.5167/uzh-21708

Download

[img]
Preview
Filetype: PDF
Size: 1MB
View at publisher

TrendTerms

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.
You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.

Author Collaborations