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Stochastic analysis and related topics

随机分析和相关主题

基本信息

批准号：
1405169
负责人：
Maria Gordina
金额：
$ 28.8万
依托单位：
University of Connecticut
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-06-01 至 2018-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405169&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. In particular, this research will provide a better understanding of Gaussian-type measures on infinite-dimensional curved spaces. The proposed research is motivated by physics. For example, infinite-dimensional spaces such as loop groups and path spaces appear in quantum field theory (QFT). The PI proposes to formalize and study some of the notions used in physics, such as measures on certain infinite-dimensional spaces. For example, it is common to see computations in QFT literature involving integrals over infinite-dimensional spaces with respect to a fictitious infinite-dimensional Lebesgue measure with a Gaussian density normalized by a constant which is infinite. Mathematically this measure can be interpreted as a Wiener measure on a flat space, or as a heat kernel measure on an infinite-dimensional curved space. In addition, this research will connect diverse fields: stochastic analysis, geometric analysis, representation theory and mathematical physics. This project is focused on elliptic and subelliptic diffusions 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. In general these infinite-dimensional spaces do not have an analogue of the Lebesgue measure or a Haar measure in the group case. In addition, geometry of these spaces will be studied in connection with smoothness properties of heat kernel measures in both elliptic (Riemannian) and subelliptic (sub-Riemannian) settings. The smoothness is interpreted as a Cameron-Martin type quasi-invariance. 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 of Brownian and energy representations of path groups. In addition, smoothness of finite-dimensional hypo-elliptic heat kernels will be studied. This project will use new techniques coming from harmonic analysis on such spaces. The educational component of the proposal is manifold: a number of projects involve graduate students of the PI; in addition the PI is involved in mentoring and educational activities from the middle school to postdoc levels.

这个项目致力于研究无限维的随机分析。特别是，本研究将提供对无限维弯曲空间上的高斯测度的更好理解。这项拟议的研究是由物理学推动的。例如，无限维空间如环路群和路径空间出现在量子场论（QFT）中。PI建议形式化和研究一些物理学中使用的概念，例如在某些无限维空间上的度量。例如，在QFT文献中经常看到计算涉及无限维空间上对一个虚构的无限维勒贝格测度的积分，该测度的高斯密度被一个无穷大的常数归一化。数学上这个测度可以解释为平坦空间上的维纳测度，或者是无限维弯曲空间上的热核测度。此外，这项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。本课题主要研究无限维弯曲空间中的椭圆型和亚椭圆型扩散，如无限维群、环路群和路径空间。研究了该方程解的存在唯一性和解的光滑性问题。一般来说，这些无限维空间在群情况下没有类似勒贝格测度或哈尔测度。此外，这些空间的几何将与椭圆（黎曼）和亚椭圆（亚黎曼）设置下的热核测度的光滑性有关。平滑性被解释为Cameron-Martin型准不变性。这本身就是一个有趣的问题，此外，它可以引起无限维群的幺正表示。该提案的一部分致力于研究路径群的布朗和能量表示。此外，还研究了有限维准椭圆型热核的光滑性。这个项目将使用来自谐波分析的新技术来分析这些空间。该提案的教育成分是多方面的：一些项目涉及PI的研究生；此外，PI还参与了从中学到博士后水平的指导和教育活动。