Infinite-dimensional stochastic analysis

本项目致力于研究无限维的随机分析。主要研究无限维空间中的随机微分方程，如无限维群、循环群、路空间、非交换空间等。本文研究了这类方程解的存在唯一性和解的光滑性问题。这些解决方案将用于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），如无限维海森堡群和Virasoro群。一般来说，这些无限维空间没有类似的勒贝格测度或哈尔测度在组的情况下。本文研究了这些测度的Cameron-Martin型拟不变性。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。本文提出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，其智力价值在于更好地理解无限维曲空间上的Gaussian型测度.特别是，拟议的研究将连接不同的领域：随机分析，几何分析和数学物理。这个研究项目对数学的各个领域都有广泛的影响，它涉及的活动有助于传播该领域新发现的知识。这个研究的动机是多个学科的：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提出形式化和研究物理学中使用的一些概念，例如某些无限维空间上的测度。此外，它还具有重要的教育内容，即涉及两名PI研究生。

This project is devoted to the study of stochastic analysis ininfinite dimensions. The main topic is stochastic differentialequations (SDEs) in infinite-dimensional spaces, such asinfinite-dimensional groups, loop groups and path spaces,non-commutative $L^p$-spaces. The questions of existence anduniqueness of solutions of the SDEs and smoothness of solutions willbe studied. These solutions will be used to construct and study heatkernel measures (a non-commutative analogue of Gaussian or Wienermeasure) on infinite-dimensional manifolds such as aninfinite-dimensional Heisenberg group and the Virasoro group. Ingeneral these infinite-dimensional spaces do not have an analogue ofthe Lebesgue measure or a Haar measure in the group case. The PIintends to study Cameron-Martin type quasi-invariance of thesemeasures. It is an interesting question in itself, and in addition itcan give rise to unitary representations of the infinite-dimensionalgroups. It is proposed to study properties of square-integrableholomorphic functions, including non-linear analogues of theSegal-Bargmann transform and bosonic Fock space representations.The intellectual merit of this proposal is in providing a betterunderstanding of Gaussian-type measures on infinite-dimensionalcurved spaces. In particular, the proposed research will connectdiverse fields: stochastic analysis, geometric analysis andmathematical physics. This research project has broader impacts ondiverse areas of mathematics, and it involves activities which helpto disseminate the knowledge of new findings in the field. Theproposed research is motivated by several subjects.Infinite-dimensional spaces such as loop groups and path spacesappear in physics, for example, in quantum field theory and stringtheory. The PI proposes to formalize and study some of the notionsused in physics, such as measures on certain infinite-dimensionalspaces. In addition, it has a significant educational component,namely, it involves two graduate students of the PI.