Stochastic Averaging: Geometry and Stratified Spaces

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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.