Stochastic Analysis and Related Topics

0407819Hsu，这项建议的主要研究者(PI)计划研究黎曼流形上的路和圈空间的各种解析和随机性质。其目标是通过提出和解决新的问题，在这一数学领域开辟新的天地，同时为进一步的研究设计一个实质性的研究计划。黎曼流形上的路和圈空间自然具备分析理论的基本成分，即Wiener测度和可微结构。这些都是重要的无限维数学和物理模型。PI将研究与这些模型有关的下列问题：(1)流形与Neumann边界条件的分部积分公式(绝热情况)；(2)单连通流形的对数Sobolv及相关的不等式；(3)流形上布朗运动的耦合，特别是最大耦合问题；(4)与Wiener测度相关的无限维质量传输问题。数学对象的重要性，例如黎曼流形上的路和圈空间，已经被数学家和物理学家认识到很长一段时间了。对他们的研究的需要是由应用数学以及理论和应用物理中的各种问题推动的。从数学和物理的角度更好地理解这些模型将有助于我们目前在这两个领域的知识。这些研究的实际方面将对数学和物理科学的未来发展具有重要意义。

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.