Mathematical Sciences: Stochastic Analysis on Manifolds

9626142李雪梅摘要本文的目的是研究流形上的随机流，包括它的存在性、几何性质及其在研究偏微分方程、非线性方程、受白噪声扰动的哈密顿系统以及基础空间的几何和拓扑等方面的应用。其中一个要研究的对象是与随机微分方程有关的仿射联系，这将在许多方向上成为有用的工具。这种联系为获得随机微分方程导数流的良好估计提供了几何结构。特别研究了由退化系统产生的结构：存在定义问题和奇点的存在问题。虽然主要的工作是关于有限维系统，但也考虑了无限维系统。本文还讨论了与之相关的路径空间上的Bismut型公式和分部积分公式。在更直接的几何水平上，我们考虑了曲率的谱正性的概念及其对流形几何的影响。随机流通常是随机微分方程组(受白噪声干扰的方程)的解。随机流的性质，包括稳定性和不变性，当与底层空间的几何一起考虑时，特别有趣。文中所描述的技术被用于研究非线性方程，包括解的爆破和正则性。研究人员还研究了局部鞅的性质，发现它在数学金融中的应用。

9626142 Li, Xue-Mei ABSTRACT The aim of this proposal is to study stochastic flows on manifolds, including their existence, geometric properties and applications to the study of partial differential equations, non-linear equations, Hamiltonian systems perturbed by white noise and the geometry and topology of the underlying space. One of the objects to study, which will be a useful tool in many directions, is the affine connection associated to a stochastic differential equation. This connection provides the geometrical structure to obtain good estimates for the derivative flow of stochastic differential equations. Those structures arising from degenerate systems are particularly investigated: there are problems of definition and of the existence of singularities. Although the main work is on finite dimensional systems infinite dimensional ones are also considered. Bismut type formulae and integration by parts formulae on path spaces connected to the above are also examined. On a more directly geometrical level the concepts of spectral positivity for curvatures are considered together with its consequences for the geometry of the manifold. Stochastic flows are typically solutions of stochastic differential equations (equations perturbed by white noise). Properties of stochastic flows including the stability and invariance are particularly interesting when considered together with the geometry of the underlying spaces. The techniques described in the proposal are used to study nonlinear equations including the blow ups and regularities of the solutions. The investigator also examines properties of local martingales, which finds its applications in mathematical finance.