RUI: Problems in Stochastic Geometry

这位研究者计划研究随机微分几何领域的三个问题。第一个问题涉及Gauss-Bonnet-Chern定理。证明了这一定理的推广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。调查者希望简化他的证明，用更基本和更透明的随机论证取代分裂原理的使用，这是他论证中的关键一步。第二个问题是关于紧流形上扩散过程定律的分部积分公式的推导。研究者最近发现了一种在扩散是严格椭圆的情况下获得分部积分公式的新方法。他计划用他的方法研究退化(非椭圆)扩散的类似问题。他相信他将能够证明一个二分法定理，对于退化情况下的一大类向量场，它给出了非常接近于按部分积分公式存在的充要条件。在第三个问题中，研究者将与Elton Hsu合作，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。他们将利用从这项工作中获得的专业知识来研究带边界的自旋流形上的狄拉克算子的指数定理。该提议结合了两个数学领域，随机分析，研究随机性，以及微分几何，研究形状。这两个领域都与物质世界有着密切的联系。例如，作为该提案的中心主题的随机微分方程被广泛用于对受随机噪声影响的物理系统进行建模。例如，材料中的热流、天气系统、航天器的轨迹以及股票期权的定价。曲线空间通常是这些现象的自然环境，例如，如果有人想要研究圆柱形管道上不同点处的热量。深入理解物理系统背后的数学原理，往往是建立这些系统的良好数学模型的关键。因此，尽管这是一项理论性质的研究，但本提案中概述的研究可能在应用科学和技术的许多领域有用。

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.