Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime
| Autor: | Michael Eckhoff |
|---|---|
| Přispěvatelé: | University of Zurich, Eckhoff, M |
| Rok vydání: | 2005 |
| Předmět: |
semiclassical limit
Witten’s Laplace Statistics and Probability ground-state splitting ground Inverse Reversible diffusion 60J60 35P20 (Primary) 31C15 31C05 35P15 58J50 58J37 60F10 60F05. (Secondary) 58J50 large deviations metastability potential theory 510 Mathematics reversibility Mathematics::Probability 35P20 31C05 60F05 exit problem FOS: Mathematics exponential distribution 1804 Statistics Probability and Uncertainty Nabla symbol 2613 Statistics and Probability Connection (algebraic framework) relaxation time Perron–Frobenius eigenvalues Brownian motion 60J60 Mathematical physics Mathematics Capacity Generator (category theory) state splitting Probability (math.PR) Zero (complex analysis) Order (ring theory) 10123 Institute of Mathematics 58J37 31C15 eigenvalue problem Statistics Probability and Uncertainty diffusion process Mathematics - Probability 35P15 60F10 |
| Zdroj: | Ann. Probab. 33, no. 1 (2005), 244-299 |
| ISSN: | 0091-1798 |
| Popis: | 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+\sqrt2\epsilon dW_t,\qquad \epsilon >0, and the spectrum near zero of its generator -L_{\epsilon}\equiv \epsilon \Delta -\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 \epsilon \downarrow 0 with optimal and uniform error estimates. Each metastable state can be viewed as an eigenstate of L_{\epsilon} with eigenvalue which converges to zero exponentially fast in 1/\epsilon. Modulo errors of exponentially small order in 1/\epsilon 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. Comment: Published 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) |
| Databáze: | OpenAIRE |
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