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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.

该项目致力于无限维随机分析的研究。特别是，这项研究将提供对无限维弯曲空间上的高斯型测度的更好理解。拟议的研究是由物理学推动的。例如，循环群和路径空间等无限维空间出现在量子场论（QFT）中。 PI 建议形式化并研究物理学中使用的一些概念，例如某些无限维空间的测量。例如，在 QFT 文献中常见的计算涉及无限维空间上的积分，涉及虚构的无限维勒贝格测度，其高斯密度由无限常数归一化。从数学上讲，该测度可以解释为平坦空间上的维纳测度，或者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、表示论和数学物理。该项目主要研究无限维弯曲空间中的椭圆和亚椭圆扩散，例如无限维群、环群和路径空间。研究SDE解的存在性、唯一性以及解的平滑性问题。一般来说，这些无限维空间没有群情况下的勒贝格测度或哈尔测度的类似物。此外，还将结合椭圆（黎曼）和次椭圆（亚黎曼）设置中热核测度的平滑特性来研究这些空间的几何形状。平滑度被解释为卡梅伦-马丁型准不变性。这本身就是一个有趣的问题，此外它还可以产生无限维群的统一表示。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，还将研究有限维亚椭圆热核的光滑度。该项目将使用来自此类空间的调和分析的新技术。该提案的教育部分是多方面的：许多项目涉及 PI 的研究生；此外，PI 还参与从中学到博士后级别的指导和教育活动。