Large deviations and a Kramers' type law for self-stabilizing diffusions

• 成果类型: Article
• 署名作者: Herrmann, Samuel; Imkeller, Peter; Peithmann, Dierk
• 署名单位: Universite de Lorraine; Humboldt University of Berlin
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/07-AAP489
• 发表日期: 2008
• 页码: 1379-1423
• 关键词: nonlinear parabolic equations; invariant probability

摘要

We investigate exit times from domains of attraction for the motion of a self-stabilized particle traveling in a geometric (potential type) landscape and perturbed by Brownian noise of small amplitude. Self-stabilization is the effect of including an ensemble-average attraction in addition to the usual state-dependent drift, where the particle is supposed to be suspended in a large population of identical ones. A Kramers' type law for the particle's exit from the potential's domains of attraction and a large deviations principle for the self-stabilizing diffusion are proved. It turns out that the exit law for the self-stabilizing diffusion coincides with the exit law of a potential diffusion without self-stabilization and a drift component perturbed by average attraction. We show that self-stabilization may substantially delay the exit from domains of attraction, and that the exit location may be completely different.