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，包括其存在性、几何性质以及在偏微分方程、非线性方程、受白色噪声扰动的Hamilton系统以及底层空间的几何和拓扑研究中的应用。随机微分方程的仿射联络是研究的对象之一，它在许多方面都是一个有用的工具。这种连接提供了几何结构，以获得良好的估计的衍生流的随机微分方程。 这些结构所产生的退化系统进行了特别调查：有定义和奇点的存在问题。虽然主要工作是有限维系统无限维的也被认为是。铋型公式和积分的部分公式的道路空间连接到上述也检查。在一个更直接的几何水平上的概念的谱正曲率被认为是连同其后果的几何形状的流形。 随机流是随机微分方程（受白色噪声扰动的方程）的典型解。随机流的性质，包括稳定性和不变性是特别有趣的，当考虑与几何的基础空间。 在建议中描述的技术被用来研究非线性方程，包括爆破和爆破的解决方案。研究者还考察了局部鞅的性质，发现它在金融数学中的应用。

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.