Infinite-dimensional stochastic analysis

本项目致力于无限维随机分析的研究。主要研究无限维空间中的随机微分方程(SDE)，如无限维群、循环群、路空间、非对易$L p-空间等。研究了随机微分方程解的存在唯一性和解的光滑性问题。这些解将被用来构造和研究无限维流形上的热核测度(高斯或维纳测度的非对易类似)，如无限维海森堡群和Virasoro群。一般说来，这些无限维空间没有类似于群情况下的勒贝格测度或Haar测度。PI打算研究这些度量的卡梅隆-马丁型准不变性。这本身就是一个有趣的问题，而且它还可以给出无限维群的么正表示。本文提出研究平方可积全纯函数的性质，包括西格尔-巴格曼变换和玻色子Fock空间表示的非线性模拟。这一建议的学术价值在于更好地理解了无限维曲空间上的高斯型测度。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。这项研究项目对数学的不同领域有更广泛的影响，它涉及有助于传播该领域新发现的知识的活动。提出的研究是由几个主题推动的。无限维空间，如环群和路径空间，出现在物理中，例如在量子场论和弦理论中。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，它还有一个重要的教育组成部分，即它涉及两名国际和平研究所的研究生。

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.