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

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dc.creatorEckhoff, Michael
dc.date2005-03-25
dc.date.accessioned2026-07-07T05:18:29Z
dc.date.available2026-07-07T05:18:29Z
dc.descriptionWe investigate the close connection between metastability of the reversible diffusion process X defined by the stochastic differential equation dX_t=-\nabla F(X_t) dt+\sqrt2εdW_t,\qquad ε>0, and the spectrum near zero of its generator -L_ε\equiv εΔ-\nabla F\cdot\nabla, where F:R^d\to R and W denotes Brownian motion on R^d. 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 ε\downarrow 0 with optimal and uniform error estimates. Each metastable state can be viewed as an eigenstate of L_ε with eigenvalue which converges to zero exponentially fast in 1/ε. Modulo errors of exponentially small order in 1/εthis 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.
dc.descriptionPublished at http://dx.doi.org/10.1214/009117904000000991 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
dc.identifierhttps://arxiv.org/abs/math/0503600
dc.identifierhttp://arxiv.org/abs/math/0503600
dc.identifierAnnals of Probability 2005, Vol. 33, No. 1, 244-299
dc.identifierdoi:10.1214/009117904000000991
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/74679
dc.subjectProbability
dc.subject60J60, 35P20 (Primary) 31C15, 31C05, 35P15, 58J50, 58J37, 60F10, 60F05. (Secondary)
dc.titlePrecise asymptotics of small eigenvalues of reversible diffusions in the metastable regime
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