Mathematical Sciences: Investigations in Probability and Erodic Theory

数学科学：概率研究和侵蚀理论

• 批准号: 9005118
• 负责人: Steven Lalley
• 金额: $ 10.08万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-06-15 至 1993-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9005118&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Investigations; Probability; Erodic

This research will be conducted in several areas of probability and ergodic theory: 1) Construction and fundamental properties of Gibbs states on sequence spaces with countable alphabets; 2) applications to dynamical systems with countable symbolic dynamics, e.g., geodesic flows on noncompact manifolds and conformal dynamical systems with parabolic points; 3) related problems concerning statistical regularities of orbits in flows; and 4) temporal and spatial growth in branching diffusion processes and branching random walks. This research is a blend of statistical theory, dynamical systems, and ergodic theory. There is a broad class of dynamical systems for which the well-developed machinery of symbolic dynamics is available. This class is characterized by the fact that the associated symbolic dynamics needs only a finite alphabet (so one has to deal with finite state Markov processes). This research extends this class to those systems for which an infinitely countable alphabet becomes necessary and develops the corresponding machinery. This will provide the tools for obtaining new results about the clustering of orbits at limit sets and about the statistics of closed geodesics on Riemannian manifolds. The outcome of this work will be to unite many known results which were won by very different methods, so unifying, thus simplifying, areas with quite separate results.

这项研究将在以下几个领域进行： 概率和遍历理论：1）构造和 序列上Gibbs态的基本性质 空间与可数字母表; 2）应用到 具有可数符号动力学的动力系统， 例如，在一个实施例中，非紧流形上的测地线流， 具有抛物点的共形动力系统; 3） 统计学中的有关问题 流中的轨道;以及4） 分支扩散过程和分支随机过程 走路. 这项研究融合了统计理论， 动力系统和遍历理论。 有一个广泛 一类动力系统，其发展良好的 符号动力学的机制是可用的。 这个类 其特征在于， 符号动力学只需要一个有限的字母表（所以一个 必须处理有限状态马尔可夫过程）。 这 研究将这类系统扩展到了 一个无限可数的字母表成为必要， 开发相应的机器。 这将提供 用于获得有关聚类的新结果的工具 关于极限集的轨道和 黎曼流形上的闭测地线 成果 这项工作的一部分将是把许多已知的结果结合起来， 都是以不同的方式赢得的，如此统一， 简化后，各个领域的结果截然不同。