圆映射的高斯扰动：谱方法
Gaussian perturbations of circle maps: A spectral approach
摘要：
本文研究了圆上离散时间动力系统的高斯扰动所对应的马尔可夫转移算子的谱性质。我们开发了一种方法，用于计算渐近表达式的特征值（在零噪声限制），并表明，变化的数量或周期的稳定轨道的确定性系统对应的变化的数量限制模1特征值的过渡算子的扰动过程。我们称这种现象为$λ$ -分岔。相应的本征函数和本征测度的渐近表达式也推导出来，并与厄米函数。
英文摘要：
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 $λ$-bifurcation. Asymptotic expressions for the corresponding eigenfunctions and eigenmeasures are also derived and are related to Hermite functions.
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