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

• 成果类型: Article
• 署名作者: Eckhoff, M
• 署名单位: University of Zurich
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000991
• 发表日期: 2005
• 页码: 244-299
• 关键词: semi-classical limit; Stochastic dynamics; markov semigroups; multiple wells; BEHAVIOR; operator; probabilities; SPECTRA; systems; times

摘要

We investigate the close connection between metastability of the reversible diffusion process X defined by the stochastic differential equation dX(t) = -delF(X-t) dt + root2epsilondW(t), epsilon > 0, and the spectrum near zero of its generator -L-epsilon equivalent to epsilonDelta - delF (.) del, 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 resealed relaxation time converges to the exponential distribution as epsilon down arrow 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.