RUI: Problems in Stochastic Geometry

RUI：随机几何问题

• 批准号: 0451194
• 负责人: Denis Bell
• 金额: --
• 依托单位: University of North Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-11-01 至 2009-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0451194&HistoricalAwards=false
• 关键词: RUI; Problems; Stochastic; Geometry

The investigator plans to study three problems in the area of stochasticdifferential geometry. The first problem concerns the Gauss-Bonnet-Cherntheorem. The investigator has proved a generalization of this theorem validfor an oriented even-dimensional Riemannian vector bundle over a closedcompact manifold, equipped with a metric connection. The investigator wouldlike to simplify his proof, replacing the use of the Splitting Principle,which constitutes a key step in his argument, with a more elementary andtransparent stochastic argument. The second problem concerns the derivationof integration by parts formulae for the law of a diffusion process on acompact manifold. The investigator has recently found a new method forobtaining integration by parts formulae in the case where the diffusion isstrictly elliptic. He plans to use his method to study the analogous problemfor degenerate (non-elliptic) diffusions. He believes he will be able toprove a dichotomy theorem giving conditions very close to necessary andsufficient for the existence of integration by parts formulae, for a largeclass of vector fields in the degenerate case. In the third problem, theinvestigator in collaboration with Elton Hsu, will use ideas in two papersof Hsu to give a simplified stochastic proof of the Gauss-Bonnet-Cherntheorem for manifolds with boundary. They will use the expertise they gainfrom this work to study the index theorem for the Dirac operator on spinmanifolds with boundary. The proposal combines two areas of mathematics, stochastic analysis, thestudy of randomness, and differential geometry, the study of shape. Bothareas have close connections with the physical world. For example,stochastic differential equations, a central theme of the proposal, arewidely used to model physical systems subject to the influence of randomnoise. Examples include the flow of heat in a material, weather systems,the trajectory of a spacecraft, and the pricing of stock options. Curvedspaces are often the natural setting for these phenomena, e.g. if one wishesto study the heat at various points on a cylindrical pipe. A deeperunderstanding of the mathematics underlying physical systems is oftencrucial in developing good mathematical models of these systems. Thus,although it is of a theoretical nature, the research outlined in thisproposal may prove useful in many areas of applied science and technology.

研究者计划研究随机微分几何领域的三个问题.第一个问题是关于高斯-博内-陈恩定理。研究者证明了这个定理的一个推广，它对具有度量联络的闭紧流形上的定向偶维黎曼向量丛是有效的。研究者想简化他的证明，用一个更基本和透明的随机论证来代替分裂原理的使用，这是他论证中的一个关键步骤。第二个问题是紧流形上扩散过程的分部积分公式的推导。研究者最近发现了一种新的方法，在扩散是严格椭圆的情况下获得分部积分公式。他计划用他的方法来研究退化（非椭圆）扩散的类似问题。 他相信，他将能够证明二分法定理给予的条件非常接近必要和充分的存在整合的部分公式，为一大类向量场的退化情况。 在第三个问题中，研究者与Elton Hsu合作，将利用Hsu的两篇论文中的思想给出带边界流形的Gauss-Bonnet-Chern定理的简化随机证明。他们将使用他们从这项工作中获得的专业知识来研究具有边界的自旋流形上的狄拉克算子的指数定理。 该建议结合了两个领域的数学，随机分析，随机性的研究，和微分几何，形状的研究。这两个领域都与物理世界有着密切的联系。例如，随机微分方程，该提案的中心主题，被广泛用于模拟受随机噪声影响的物理系统。 例子包括材料中的热流、天气系统、航天器的轨迹和股票期权的定价。弯曲空间通常是这些现象的自然环境，例如，如果一个人希望研究圆柱形管道上不同点的热量。深入理解物理系统的数学基础对于建立这些系统的良好数学模型通常是至关重要的。因此，虽然它是一个理论性的，在这一建议概述的研究可能会证明有用的应用科学和技术的许多领域。