Mathematical Sciences: Precise path analysis of Brownian motions on non-smooth Riemannian manifolds

数学科学：非光滑黎曼流形上布朗运动的精确路径分析

• 批准号: 9204038
• 负责人: Weian Zheng
• 金额: $ 4.5万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1995-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204038&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Precise; path; analysis

The proposed research is related to several areas of probability theory: the theory of symmetric diffusion processes and Dirichlet forms, stochastic differential geometry, and the analysis of heat kernels. The principal investigator plans to extend the results of Carmona and Zheng about the small time asymptotics of the probability that Brownian paths visiting the sets of a given finite sequence. These extensions will be used to solve several questions: the monotonicity of heat kernels associated to Newmann boundary conditions (Chavel's conjecture) for all times, the study of Riemannian manifolds with piecewise constant metrics via the precise path behavior of Brownian motions on these manifolds, and the study of general non-smooth Riemannian manifolds and in particular the Harnack's inequality by using the above concrete results and weak convergence. There have been a lot of studies about the propagation of heat on some geometric body with smooth surfaces made by some unitary material. The situation becomes more complicated when the surface is not smooth or the body is made with materials of different thermal conductivities. This research is aimed at a microscopic analysis of the latter case using probability theory.

拟议的研究涉及以下几个领域： 概率论：对称扩散过程理论 和狄利克雷形式，随机微分几何， 热核分析 首席研究员计划 推广了Carmona和Zheng关于小时间的结果 布朗路径的概率渐近性 给定有限序列的集合。 这些扩展将用于 解决了几个问题：热核的单调性 Newmann边界条件（Chavel猜想） 对于所有的时间，研究黎曼流形与分段 通过布朗的精确路径行为的常数度量 这些流形上的运动，以及一般非光滑的研究 黎曼流形，特别是Harnack不等式 利用上述具体结果和弱收敛性， 有很多关于传播的研究 在具有光滑表面几何体上加热， 单一材料 情况变得更加复杂， 表面不光滑或主体由以下材料制成 不同的导热系数。 这项研究旨在 用概率论对后一种情况进行微观分析。