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.
