Stochastic analysis in infinite dimensions

这个项目致力于研究无限维的随机分析。本文主要研究无限维空间中的随机微分方程（SDEs），如非交换L^p空间、相关无限维群、环路群和路径空间。研究了该方程解的存在唯一性和光滑性问题。然后将解用于构造和研究无限维流形上的热核测度（高斯测度或维纳测度的非交换模拟）。一般来说，这些无限维群不是局部紧化的，因此没有哈尔测度。PI打算研究这些测度的Cameron-Martin型拟不变性。这本身就是一个有趣的问题，但它也可以引起无限维群的幺正表示。提出研究平方可积全纯函数的性质。例如，准不变性可以用来证明全纯函数的弱Cauchy-Riemann方程。除了经典的无限维随机分析外，PI还打算研究非交换的SDEs。拟议的研究是由几个主题推动的。无限维空间，如环路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，例如在某些无限维空间上的度量。非交换概率领域提出的问题起源于量子物理学。

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.