Stochastic Analysis in Differential Geometry

微分几何中的随机分析

• 批准号: 0104079
• 负责人: Elton Hsu
• 金额: $ 9.5万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104079&HistoricalAwards=false
• 关键词: Stochastic; Analysis; Differential; Geometry

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.

研究者将研究与有限和无限微分几何相关的各种分析和概率问题，特别是具有边界和路径的黎曼流形以及此类流形上的循环空间。在黎曼流形与边界的情况下，他将试图获得一个有用的公式费曼-卡茨公式的向量丛（主要是微分形式），可以有效地用于一些问题。这类公式以前曾得到过，但不容易应用于这些问题。他将证明热半群的导数可以根据Ricci曲率和边界的第二基本形式显式有界。他将采用的方法为有限维问题是密切相关的随机分析路径空间的黎曼流形。在无限维的设置，他希望打破新的道路和循环空间分析的情况下，研究流形有一个边界，从而开辟了新天地。具体来说，他将试图证明一个集成的部分公式在路径空间的这种情况下。这是研究有边界流形的合理起点，因为众所周知，分部积分公式是路径和回路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。在这一领域研究的许多模型（例如，路径和回路空间）是物理学和其他相关科学和工程领域（特别是高能物理学和空间技术）中具体模型的数学抽象。了解这些模型的数学结构通常是走向实际应用的第一步。