Stochastic Averaging: Geometry and Stratified Spaces

本研究主要围绕随机平均模型降阶技术中的一些问题展开。虽然随机平均法已经存在了至少30年，但最近随机分析的发展提出了一种新的观点，特别是现在很清楚，简化模型可以在分层空间中取值。该建议概述了一些问题，目标是更好地理解分层空间上的随机平均和马尔可夫过程。 粗略地说，拟议的研究试图更好地了解小噪声对振荡的影响。由于许多机械、制造和生物系统都具有受小噪声影响的振荡行为，因此所提出的研究可以为我们对许多系统的理解和设计提供信息。这项研究的更具体的目标是找到简化更复杂的模型的准确方法。这些简化的模型可用于控制和设计程序。

This research centers around some problems in the technique of model reduction known as stochastic averaging. Although stochastic averaging has been around for at least 30 years, recent developments in stochastic analysis suggest a new look at it. In particular, it is now clear that the reduced model can take values in a stratified space. This proposal outlines a number of questions, with the goal of a better understanding of both stochastic averaging and Markov processes on stratified spaces. Roughly, the proposed research attempts to better understand the effects of small noise upon oscillations. As many mechanical, manufacturing, and biological systems have oscillatory behavior affected by small noise, the proposed research can inform our understanding and design of a host of systems. The more particular goal of this research is to find accurate methods of simplifying more complicated models. These simplified models could then be used in control and design procedures.