Stochastic Analysis on Manifolds

流形的随机分析

• 批准号: 0072387
• 负责人: Xue-Mei Li
• 金额: $ 4.8万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-15 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072387&HistoricalAwards=false
• 关键词: Stochastic; Analysis; Manifolds

The proposed research falls into the following broad areas of stochastic analysis: stochastic dynamical systems on manifold, the geometry of stochastic flows and Hormander type second order differential operators, analysis on path spaces, and the interplay between the dynamical behavior of stochastic dynamical systems and the geometry and topology of the underlying spaces. The investigator shall concentrate on (1) L^2 theory for differential forms on path and loop spaces; (2) linear connections associated to Hormander type second order elliptic operators; (3) stochastic flows and in particular stochastic differential equations associated to Hamiltonian systems. The study of lattice models with spin variable in a manifold is also included, as is the study of bounded and L^2 harmonic forms on finite dimensional manifolds. The investigator will study these problems using techniques from stochastic differential equations. This work would lead to a better understanding of random perturbation of dynamical systems and geometric analysis of infinite dimensional spaces. The proposed project concentrates in the areas of stochastic dynamical systems and analysis on path spaces. Path spaces are the spaces of trajectories of particles or other physical objects. Methods from stochastic analysis, especially the techniques arising from the study of stochastic differential equations, shall be employed. Stochastic differential equations are themselves very important objects to study as they arise naturally from physics, engineering, biology, and finance, to model the influence of white noise. The study of such equations on a variety of geometric objects such as spheres in connection with the curvature of such spaces arises naturally from the mathematics and also in applications, for example in the computation of recovering data by filtering out the white noise in the transmission.

福尔斯的研究主要集中在以下几个方面：流形上的随机动力系统、随机流的几何和Hormander型二阶微分算子、路空间的分析以及随机动力系统的动力学行为与其底层空间的几何和拓扑之间的相互作用。研究者将集中于（1）L^2理论的微分形式的道路和循环空间;（2）线性连接相关的霍曼德型二阶椭圆算子;（3）随机流，特别是随机微分方程相关的哈密顿系统。也包括流形中自旋变量的格子模型的研究，以及有限维流形上有界和L^2调和形式的研究。调查员将研究这些问题使用随机微分方程的技术。这项工作将导致更好地了解随机扰动的动力系统和几何分析的无限维空间。拟议的项目集中在随机动力系统和路径空间分析领域。路径空间是粒子或其他物理对象的轨迹空间。应采用随机分析方法，特别是随机微分方程研究中产生的技术。随机微分方程本身是非常重要的研究对象，因为它们自然地出现在物理学、工程学、生物学和金融学中，用于模拟白色噪声的影响。对各种几何对象（例如与这种空间的曲率有关的球体）上的这种方程的研究自然地从数学中产生，并且也在应用中产生，例如在通过过滤掉传输中的白色噪声来恢复数据的计算中。