AVERAGING OF HAMILTONIAN FLOWS WITH AN ERGODIC COMPONENT

• 成果类型: Article
• 署名作者: Dolgopyat, Dmitry; Koralov, Leonid
• 署名单位: University System of Maryland; University of Maryland College Park
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/07-AOP372
• 发表日期: 2008
• 页码: 1999-2049
• 关键词: random perturbations

摘要

We consider a process oil 72, which consists of fast motion along the stream lines of ail incompressible periodic vector field perturbed by white noise. It gives rise to a process on the graph naturally associated to the structure of the stream lines Of the unperturbed flow. It has been shown by Freidlin and Wentzell [Random Perturbations of Dynamical Systems, 2nd ed. Springer. New York (1998)] and [Mem. Amer Math. Soc. 109 (1994)] that if the stream function of the flow is periodic. then the corresponding process oil the graph weakly converges to a Markov process. We consider the situation where the stream function is not periodic, and the How (when considered on the torus) has an ergodic component of positive measure. We show that if the rotation number is Diophantine. then the process oil the graph still converges to a Markov process, which spends I positive proportion of time in the vertex corresponding to the ergodic component of the flow.