Dynamical Theory of Stochastic Flows

小行星9703693 本计画系关于随机流产生之动力学理论 流形上的随机微分方程PI计划调查以下内容 问题 它已被证明在一些例子中，一个随机流可以被分解 作为一个渐进确定性的流动，随后是一个随机的“旋转”。PI计划 研究一些一般类型的随机流具有这种 分解如果理解了这一点，那么一个随机源和汇的理论就可以 为随机流而开发。它已经建立了随机流包含在 有限维李变换群，随机流的极限稳定性是 完全由群体的结构决定。PI计划调查什么 这对于一般的无限维随机流是成立的。 动力学理论研究系统的运动，无论是机械的， 生物或环境系统。一个重要的问题是， 稳定或系统表现出什么样的长期行为。随机流提供了 运动依赖于某些随机元素的系统的模型。虽然人们可能 无法确定地预测随机系统的路径，研究随机系统的 流程使人们能够预测系统的长期行为或模式， 信心这个建议涉及到动力学理论中的一些基本问题 随机流。

9703693 Liao This project is concerned with the dynamical theory of stochastic flows generated by stochastic differential equations on manifolds. The PI plans to investigate the following problems. It has been shown in some examples that a stochastic flow may be decomposed as an asymptotically deterministic flow followed by a random "rotation". The PI plans to look into the possibility that some general type of stochastic flows possess this kind of decomposition. If this is understood, then a theory of random sources and sinks can be developed for stochastic flows. It has been established for stochastic flows contained in finite dimensional Lie transformation groups that the limiting stability of stochastic flows is completely determined by the structure of the group. The PI plans to investigate to what extent this holds for general infinite dimensional stochastic flows. The dymanical theory studies the motion of a system, whether this is a mechanical, a biological or an environmental system. An important question is whether the system is stable or what kind of long time behavior the system exhibits. The stochastic flows provide models for systems whose motions depend on some random elements. Although one may not be able to predict with certainty the path of a random system, the study of stochastic flows enables one to project the long time behavior or pattern of the system with confidence. The proposal is concerned with some basic problems in the dynamical theory of stochastic flows.