Stochastic Analysis and Related Topics

随机分析及相关主题

基本信息

批准号：
1007496
负责人：
Maria Gordina
金额：
$ 24万
依托单位：
University of Connecticut
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2010
资助国家：
美国
起止时间：
2010-07-01 至 2015-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007496&HistoricalAwards=false
关键词：
Stochastic Analysis Related Topics

项目摘要

This project is devoted to the study of stochastic analysis in infinite dimensions. The main topic is stochastic differential equations (SDEs) in infinite-dimensional curved spaces, such as infinite-dimensional groups, loop groups and path spaces. The questions of existence and uniqueness of solutions of the SDEs and smoothness of solutions will be studied. These solutions will be used to construct and study heat kernel measures (a non-commutative analogue of Gaussian or Wiener measure) on infinite-dimensional manifolds. In general these infinite-dimensional spaces do not have an analogue of the Lebesgue measure or a Haar measure in the group case. The PI intends to study Cameron-Martin type quasi-invariance of these measures. It is an interesting question in itself, and in addition it can give rise to unitary representations of the infinite-dimensional groups. One part of the proposal is devoted to studying an energy representation of path groups. It is proposed to study properties of square-integrable holomorphic functions, including non-linear analogues of the Segal-Bargmann transform and bosonic Fock space representations. Levy processes in Lie groups will be studied.The proposed research will connect diverse fields: stochastic analysis, geometric analysis, representation theory and mathematical physics. This research project has broader impacts on diverse areas of mathematics such as stochastic analysis, representation theory, geometric analysis etc, and it involves activities which help to disseminate the knowledge of new findings in the field. The motivation comes from several subjects.Infinite-dimensional spaces such as loop groups and path spaces appear in physics, for example, in quantum field theory. The PI proposes to formalize and study some of the notions used in physics, such as measures on certain infinite-dimensional spaces. In addition, it has a significant educational component, namely, it involves graduate students of the PI.

该项目致力于无限维度中随机分析的研究。主要主题是无限尺寸弯曲空间中的随机微分方程（SDE），例如无限维组，循环基团和路径空间。将研究存在的存在和解决方案解决方案的独特性和解决方案的平滑性问题。这些解决方案将用于构建和研究无限二维流形的热内核测量（高斯或维纳仪的非共同类似物）。通常，这些无限维空间在集体案例中没有Lebesgue度量或HAAR度量的类似物。 PI打算研究这些措施的Cameron-Martin型准变革。这本身就是一个有趣的问题，此外，它还可以产生无限差异群体的统一表示。该提案的一部分致力于研究路径组的能量表示。有人提出研究方形综合性溶性函数的性能，包括Segal-Bargmann变换和玻色子Fock空间表示的非线性类似物。将研究谎言组中的征费过程。拟议的研究将连接不同的领域：随机分析，几何分析，表示理论和数学物理学。该研究项目对数学领域（例如随机分析，表示理论，几何分析等）产生了更广泛的影响，并且涉及有助于传播该领域新发现的知识的活动。动机来自几个主题。在物理学中出现在量子场理论中，例如循环群和路径空间等二维空间。 PI建议将物理中使用的一些概念形式化和研究，例如对某些无限维空间的措施。此外，它具有重要的教育部分，即涉及PI的研究生。