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Stochastic analysis in infinite dimensions

无限维随机分析

基本信息

批准号：
0306468
负责人：
Maria Gordina
金额：
$ 9.59万
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：
2003
资助国家：
美国
起止时间：
2003-06-01 至 2007-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306468&HistoricalAwards=false
关键词：
Stochastic analysis infinite dimensions

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 spaces,such as noncommutative L^p-spaces, related infinite-dimensional groups, loop groups, and path spaces. The questions of existence and uniqueness of solutions of the SDEs and smoothnessof solutions will be studied. Then the solutions will be used to construct and study heat kernel measures (a noncommutative analogue of Gaussian or Wiener measure) on the infinite-dimensional manifolds. In general these infinite-dimensional groups are not locally compact and therefore do not have a Haar measure. The PI intends to study Cameron-Martin type quasi-invariance of these measures. It is an interesting questions in itself, but it also can give rise to unitary representations of the infinite-dimensional groups. It is proposed to study properties of square-integrable holomorphic functions. For example, quasi-invariance can be used to prove weak Cauchy-Riemann equations for holomorphic functions. Besides the classical infinite-dimensional stochastic analysis the PI intends to study noncommutative SDEs. The proposed research is motivated by several subjects. Infinite-dimensional spaces such as loop groups and path spaces appear in physics, for example, in quantum field theory and string theory. The PI proposes to formalize some of the notions used in physics, such as measures on certain infinite-dimensional spaces. The proposed problems in the field of noncommutative probability have their origins in quantum physics.

本项目致力于研究无限维中的随机分析。本课程的主要内容是无限维空间中的随机微分方程，例如非交换L^p空间、相关的无限维群、循环群和路径空间。本文将研究这类方程解的存在唯一性和光滑性问题。然后利用这些解来构造和研究无限维流形上的热核测度（Gauss测度或Wiener测度的非交换模拟）。一般来说，这些无限维群不是局部紧的，因此没有哈尔测度。PI打算研究这些措施的Cameron-Martin型准不变性。这本身是一个有趣的问题，但它也可以引起无限维群的酉表示。研究平方可积全纯函数的性质。例如，拟不变性可以用来证明全纯函数的弱柯西-黎曼方程。除了经典的无穷维随机分析外，PI还打算研究非交换随机微分方程。提出的研究是出于几个主题。无限维空间，如圈群和路径空间出现在物理学中，例如量子场论和弦理论。PI建议将物理学中使用的一些概念形式化，例如某些无限维空间上的度量。在非对易概率领域提出的问题起源于量子物理学。