具有平坦哈密顿量和多时间尺度的噪声系统
A noisy system with a flattened Hamiltonian and multiple time scales
摘要：
考虑具有时间周期漂移和扩散系数的二维弱耗散动力系统。漂移的平均值由简并哈密顿量控制，简并哈密顿量的临界点的集合具有内部。在三个时间尺度下研究了系统的动力学行为。使用鞅问题的方法和分离的时间尺度，我们平均的系统显示收敛到一个分层空间上的马尔可夫过程。该平均结合了周期系数的确定性时间平均和所得系统的随机平均。约化空间的对应层是一个双球面、一个点和一个线段。特别注意的是给域的限制发电机的描述，包括在地层满足点的胶合条件的分析。这些由时间尺度层次效应引起的胶合条件与斜布朗运动区域上的胶合条件相似，并与蜘蛛鞅的描述有关.
英文摘要：
We consider a two-dimensional weakly dissipative dynamical system with time-periodic drift and diffusion coefficients. The average of the drift is governed by a degenerate Hamiltonian whose set of critical points has an interior. The dynamics of the system is studied in the presence of three time scales. Using the martingale problem approach and separating the time scales, we average the system to show convergence to a Markov process on a stratified space. The averaging combines the deterministic time averaging of periodic coefficients, and the stochastic averaging of the resulting system. The corresponding strata of the reduced space are a two-sphere, a point and a line segment. Special attention is given to the description of the domain of the limiting generator, including the analysis of the gluing conditions at the point where the strata meet. These gluing conditions, resulting from the effects of the hierarchy of time scales, are similar to the conditions on the domain of skew Brownian motion and are related to the description of spider martingales.
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