Infinite-dimensional stochastic analysis

无限维随机分析

• 批准号: 0706784
• 负责人: Maria Gordina
• 金额: $ 21.99万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706784&HistoricalAwards=false
• 关键词: Infinite; dimensional; stochastic; analysis

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.

该项目致力于研究随机分析的无限维度。主要主题是无限维空间中的随机差异（SDE），例如芬矿二维组，循环群和路径空间，非共同$ l^p $ - 空格。存在的解决方案和解决方案平滑度的存在和直觉的问题将研究。这些解决方案将用于构建和研究无限二维流形（例如Aninfinite二维的Heisenberg grout use the Virasoro group）上的无限二维歧管上的Heatkernel度量（高斯或Wienermeasian的非共同类似物）。 Ingeneral这些无限维空间在集体案例中没有类似的Lebesgue度量或HAAR度量的类似物。 piintend用于研究cameron-martin型的emeaseures型准变革。这本身就是一个有趣的问题，此外，它还可以产生无限二维组的统一表示。 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.尤其是，拟议的研究将连接界面：随机分析，几何分析和数学物理学。该研究项目具有更广泛的影响数学领域，它涉及Helpto传播该领域新发现的知识的活动。该研究的动机是由几个主题的动机。在量子场理论和弦理论中，诸如循环群和物理中的路径空间苹果等二维空间。 PI建议将某些物理学的概念形式化和研究，例如对某些无限二维空间的措施。此外，它具有重要的教育部分，即涉及PI的两名研究生。