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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.

本项目致力于研究无限维中的随机分析。主要研究无限维曲空间中的随机微分方程，如无限维群、环群和路径空间。我们将研究这类方程解的存在唯一性和解的光滑性问题。这些解决方案将被用来构建和研究无限维流形上的热核测度（高斯或维纳测度的非交换模拟）。一般来说，这些无限维空间在群的情况下没有勒贝格测度或哈尔测度的类似物。PI打算研究这些措施的Cameron-Martin型准不变性。这本身就是一个有趣的问题，此外，它还可以引起无限维群的酉表示。建议的一部分是专门研究的能量表示的路径群。研究平方可积全纯函数的性质，包括Segal-Bargmann变换和玻色Fock空间表示的非线性类似物。李群中的Levy过程将被研究，所提出的研究将连接不同的领域：随机分析，几何分析，表示论和数学物理。该研究项目对数学的各个领域产生了更广泛的影响，如随机分析，表示论，几何分析等，它涉及有助于传播该领域新发现知识的活动。其动机来自于几个学科：无限维空间，如圈群和路径空间，出现在物理学中，如量子场论。PI建议将物理学中使用的一些概念形式化并进行研究，例如某些无限维空间上的度量。此外，它还有一个重要的教育组成部分，即它涉及PI的研究生。