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

Eckhoff, Michael

Ann. Probab., Tome 33 (2005) no. 1, p. 244-299 / Harvested from Project Euclid

Résumé

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 −L_ɛ≡ɛΔ−∇F⋅∇, where F:ℝ^d→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 ɛ↓0 with optimal and uniform error estimates. Each metastable state can be viewed as an eigenstate of L_ɛ 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.

Détail

Publié le : 2005-01-14
Classification:  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,  60J60,  35P20,  31C15,  31C05,  35P15,  58J50,  58J37,  60F10,  60F05