Stochastic Analysis in Differential Geometry

研究者将研究与有限和无限微分几何相关的各种解析和概率问题，特别是黎曼流形的边界和路径以及流形上的循环空间。在有边界的黎曼流形的情况下，他将试图获得一个有用的费曼-卡茨公式的向量束（主要是微分形式）的公式，它可以有效地用于许多问题。这种类型的公式以前得到过，但不容易应用于问题。他将证明热半群的导数可以用里奇曲率和边界的第二种基本形式明确地有界。他将采用的有限维问题的方法与黎曼流形上的路径空间的随机分析密切相关。在无限维的背景下，他希望通过研究流形有边界的情况，在路径和循环空间的分析上有所突破，从而开拓新的领域。具体来说，他将尝试在这种情况下证明路径空间中的分部积分公式。这是研究有边界流形的一个合理的起点，因为众所周知，分部积分公式是路径和循环空间中许多有趣问题的中心。几何随机分析是概率论中一个活跃的研究领域。它的目标是使用随机方法（而不是分析方法）来研究具有良好定义的几何结构的模型。在这个领域中研究的许多模型（例如，路径和环路空间）是物理和其他相关科学和工程领域（特别是高能物理和空间技术）中具体模型的数学抽象。了解这些模型的数学结构通常是其实际应用的第一步。

The investigator will study various analytic and probabilistic problems related to finite and infinite differential geometry, especially Riemannian manifolds with boundary and path and loop spaces over such manifolds. In the case of Riemannian manifold with boundary, he will try to obtain a useful formulation of the Feynman-Kac formula for vector bundles (mainly differential forms) which can be used effectively in a number of problems. A formula of this type was obtained before but is not easy to be applied to the problems. He will prove that the derivatives of the heat semigroup can be bounded explicitly in terms of the Ricci curvature and the second fundamental form of the boundary. The approach he will adopt for the finite dimensional problems is intimately related to stochastic analysis on path spaces over Riemannian manifolds. In the infinite dimensional setting, he hopes to break new ground in path and loop space analysis by studying the case when the manifold has a boundary, thus breaking new ground. Specifically he will try to prove an integration by parts formula in the path space for this case. This is a reasonable starting point for investigating manifolds with boundary, for as it is well known that integration by parts formula lies at the center of many interesting problems in path and loop spaces. Stochastic analysis in geometry is an active research area in probability theory. Its goal is to use stochastic methods (as opposed to analytic methods) to investigate models with well defined geometric structures. Many models studied in this area (e.g., path and loop spaces) are mathematical abstractions of concrete models in physics and other related areas of science and engineering (especially high energy physics and space technology). An understanding of the mathematical structure of these models is usually the first step towards their practical applications.