Stochastic Analysis and Related Topics

0407819 Hsu该提案的主要研究者（PI）计划研究黎曼流形上的路径和回路空间的各种分析和随机性质。我们的目标是通过提出和解决新的问题，在这一领域的数学开辟新的天地，并在同一时间，设计一个实质性的研究计划，进一步调查。黎曼流形上的路和圈空间自然具备分析理论的基本要素，即维纳测度和可微结构。这些都是重要的无穷维数学和物理模型。PI将研究与这些模型有关的以下问题：（1）具有Neumann边界条件（绝热情况）的流形的分部积分公式;（2）单连通流形的对数Sobolev和相关不等式;（3）流形上布朗运动的耦合，特别是最大耦合问题;（4）与Wiener测度有关的无限维质量输运问题。 数学对象的重要性，例如黎曼流形上的路径和回路空间，已经被数学家和物理学家认识了很长一段时间。对他们的研究的需要是由各种问题的动机在应用数学和理论和应用物理。从数学和物理的角度更好地理解这些模型将有助于我们在这两个领域的现有知识。这些调查的实际方面将成为重要的数学和物理科学的未来发展。

0407819Hsu The Principal Investigator (PI) of this proposal plans to study various analytic and stochastic properties of path and loop spaces over a Riemannian manifold. The goal is to break new ground in this area of mathematics by proposing and solving new problems and, at the same time, devising a substantial research program for further investigation. Path and loop spaces over a Riemannian manifold are naturally equipped with the basic ingredients for an analytic theory, namely, the Wiener measure and a differentiable structure. These are important infinite dimensional mathematical and physical models. The PI will study the following problems concerning these models: (1) Integration by parts formula for manifolds with the Neumann boundary conditions (adiabatic case); (2) Logarithmic Sobolev and related inequalities for simply-connected manifolds; (3) Couplings of Brownian motions on manifolds, especially the problem of maximal couplings; (4) Infinite dimensional mass transportation problems related to the Wiener measure. The importance of mathematical objects, such as path and loop spaces over Riemannian manifolds, has been recognized by mathematicians and physicists for quite some time. The need for their study is motivated by a variety of problems in applicable mathematics and theoretical and applied physics. A better understanding of these models from the mathematical as well as from the physical points of view will contribute to our current knowledge in these two fields. The practical aspects of these investigations will become significant in the future development of mathematical and physical sciences.