Probabilistic Methods in Geometry and Analysis

9706910许这项研究建议是首席调查员计划的一部分，该计划以解析和概率方法研究(非平坦的)有限或无限维空间(主要是黎曼流形及其上的路径和循环空间)的分析性质。该计划的PI长期目标是建立黎曼流形上的路和圈空间的完整的解析和概率理论，其对应的(平坦)欧氏空间(有限维或无限维)已经知道很长一段时间了。这些努力的成功将极大地促进我们对非线性无限维空间的普遍理解。对于分析部分，PI将把他的研究集中在进一步开发分析技术上，这些技术被归入“Malliavin微积分”的名称之下。特别是在这一建议所涉及的时期，PI将把大量的研究资源花在各种Sobolev范数的等价性(所谓的Meyer等价问题)和广义Ornstein-Uhlenbeck算子的谱性质上。在概率部分，PI打算通过样本路径对Ornstein-Uhlenbeck过程展开详细的研究，特别是曲率对过程的短时间和长时间行为的影响。正在研究的模型，如球体、椭球体和环面等几何结构上的路径和回路空间，近年来在相对论、量子物理、天体物理和宇宙学中得到了广泛的应用。首席调查员将用传统的分析方法和最新的概率论方法来研究这些模型。具体地说，他将研究特定随机过程在路径和回路空间上的长时间和短时间行为，希望它将揭示几何性质，这些性质可以被理论科学家和工程师在各自的学科中应用。

9706910 Hsu This research proposal forms an integral part of the Principal Investigator's program of studying analytical properties of (nonflat) finite or infinite dimensional spaces (mainly Riemannian manifolds and path and loop spaces over them) by analytic and probabilistic methods. The PIs long range goal for the program is to establish a complete analytical and probabilistic theory of path and loop spaces over Riemannian manifolds, whose counterpart in (flat) Euclidean space (finite or infinite dimensional) has been known for quite some time. Successes in these efforts will contribute significantly towards our general understanding of nonlinear infinite dimensional spaces. For the analytical part, the PI will concentrate his research on further developing analytic techniques subsumed under the name "Malliavin Calculus." In particular for the period covered in this proposal, the PI will spend a substantial amount of his research resources on the equivalence of various Sobolev norms (the so-called Meyer's equivalent problem) and on the spectral properties of the generalized Ornstein-Uhlenbeck operator. For the probabilistic part, the PI intends to launch a detailed study of the Ornstein-Uhlenbeck process through sample path, especially the effect of curvature on short-time and long-time behaviors of the process. The models under study, path and loop spaces over such geometric structures such as spheres, ellipsoids, and tori, have been used extensively in recent years in the theory of relativity, quantum physics, astrophysics, and cosmology. The Principal Investigator will study these models by both traditional analytical methods and the more recent methods of probability theory. Specifically he will investigate both long-time and short-time behavior of a specific random process on path and loop spaces in the hope that it will reveal geometric properties which can be applied by theoretical scientists and engineers in their respective disciplines.