The Annals of Probability

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

Michael Eckhoff

Full-text: Open access

Abstract

We investigate the close connection between metastability of the reversible diffusion process X defined by the stochastic differential equation

\[dX_{t}=-\nabla F(X_{t})\,dt+\sqrt{2\varepsilon }\,dW_{t},\qquad \varepsilon >0,\]

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.

Article information

Source
Ann. Probab., Volume 33, Number 1 (2005), 244-299.

Dates
First available in Project Euclid: 11 February 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1108141727

Digital Object Identifier
doi:10.1214/009117904000000991

Mathematical Reviews number (MathSciNet)
MR2118866

Zentralblatt MATH identifier
1098.60079

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Secondary: 31C15: Potentials and capacities 31C05: Harmonic, subharmonic, superharmonic functions 35P15: Estimation of eigenvalues, upper and lower bounds 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx] 58J37: Perturbations; asymptotics 60F10: Large deviations 60F05: Central limit and other weak theorems

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

Citation

Eckhoff, Michael. Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime. Ann. Probab. 33 (2005), no. 1, 244--299. doi:10.1214/009117904000000991. https://projecteuclid.org/euclid.aop/1108141727


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