# 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.

## 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: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics 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 English 2005 19 Feb 2010 15:48 05 Apr 2016 13:24 Institute of Mathematical Statistics 0091-1798 https://doi.org/10.1214/009117904000000991

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