Infinite-dimensional stochastic analysis

本课题致力于无限大维随机分析的研究。主要研究无穷维空间中的随机微分方程，如无穷维群、环路群、路径空间、非交换L^p -空间等。研究了该方程解的存在唯一性和解的光滑性问题。这些解将用于在无限维流形（如无限维Heisenberg群和Virasoro群）上构建和研究热核测度（高斯测度或Wienermeasure的非交换模拟）。一般来说，这些无限维空间在群情况下没有类似的勒贝格测度或哈尔测度。本文拟研究这些测度的Cameron-Martin型拟不变性。这本身就是一个有趣的问题，此外，它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，包括非线性的segal - bargmann变换和玻色子Fock空间表示。这一建议的智力价值在于提供了对无限维弯曲空间上的高斯测度的更好理解。特别是，拟议的研究将连接多个领域：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有更广泛的影响，它涉及的活动有助于传播该领域新发现的知识。提出的研究是由几个主题驱动的。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建议形式化和研究物理学中使用的一些概念，例如在某些无限维空间上的度量。此外，它有一个显著的教育成分，即，它涉及两名研究生的PI。

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.