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January 2005 Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime
Michael Eckhoff
Ann. Probab. 33(1): 244-299 (January 2005). DOI: 10.1214/009117904000000991


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.


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Michael Eckhoff. "Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime." Ann. Probab. 33 (1) 244 - 299, January 2005.


Published: January 2005
First available in Project Euclid: 11 February 2005

zbMATH: 1098.60079
MathSciNet: MR2118866
Digital Object Identifier: 10.1214/009117904000000991

Primary: 35P20 , 60J60
Secondary: 31C05 , 31C15 , 35P15 , 58J37 , 58J50 , 60F05 , 60F10

Keywords: capacity , diffusion process , eigenvalue problem , Exit problem , exponential distribution , ground-state splitting , large deviations , metastability , Perron–Frobenius eigenvalues , potential theory , relaxation time , reversibility , semiclassical limit , Witten’s Laplace

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.33 • No. 1 • January 2005
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