GAUSSIAN PERTURBATIONS OF CIRCLE MAPS: A SPECTRAL APPROACH

• 成果类型: Article
• 署名作者: Mayberry, John
• 署名单位: Cornell University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/08-AAP573
• 发表日期: 2009
• 页码: 1143-1171
• 关键词: stochastic bifurcations; oscillator

摘要

In this work, we examine spectral properties of Markov transition operators corresponding to Gaussian perturbations of discrete time dynamical systems on the circle. We develop a method for calculating asymptotic expressions for eigenvalues (in the zero noise limit) and show that changes to the number or period of stable orbits for the deterministic system correspond to changes in the number of limiting modulus 1 eigenvalues of the transition operator for the perturbed process. We call this phenomenon a A-bifurcation. Asymptotic expressions for the corresponding eigenfunctions and eigenmeasures are also derived and are related to Hermite functions.