Brownian Motion and Random Partitions

9703691皮特曼首席研究员和合著者最近的工作在随机划分、随机离散分布、随机树和布朗运动和其他随机过程的游程长度之间建立了联系。本项目继续研究这些关系及其应用。谢斯的星系引力星团模型导致了与相加聚合体有关的问题，在加性聚合体中，有限集合中的每一对集合以两个集合的大小之和的速度合并。这种可加结合的渐近性定义了一个离散测值过程，它可以通过在适当的泊松过程的点上割下Aldous的连续统随机树来构造。人们建议研究这一过程和其他模拟合并现象的度量值马尔可夫过程，并寻找这些过程的新应用。布朗运动提供了具有连续路径的连续时间随机过程的现代理论的基础，并在物理和数学金融等领域有广泛的应用。随机划分在组合学、物理学和遗传学中都有应用。聚合和破碎的物理过程的统计模型在许多领域都很有意义。粒子分成团块的演化可以有不同的解释，例如，原子分成分子，分子分成聚合物，或者恒星分成星系。在化学环境中，原子结合形成分子，分子结合形成聚合物。在宇宙学的背景下，恒星聚合形成星系。所提出的研究集中在具有有限总质量的连续体的分割成有限或可数的子质量集合的不可逆合并过程的模型，并具有质量守恒。

9703691 Pitman Recent work of the principal investigator and co-authors has made connections between random partitions, random discrete distributions, random trees, and the lengths of excursions of Brownian motion and other stochastic processes. The present project continues the study of these relations and their applications. Sheth's model for gravitational clustering of galaxies leads to problems related to the additive coalescent in which each pair of sets in a partition of a finite set is merging at a rate which is the sum of the sizes of the two sets. Asymptotics of this additive coalescent define a discrete measure valued process which can be constructed by cutting Aldous's continuum random tree at points of a suitable Poisson process. It is proposed to study this and other measure-valued Markov processes which model coalescent phenomena, and to look for novel applications of these processes. Brownian motion provides the doundation of the modern theory of continuous time random processes with continuous paths, and has applications in fields as diverse as physics and mathematical finance. Random partitions find applications to combinatorics, physics and genetics. Statistical models for physical processes of coalescence and fragmentation are of interest in a number of fields. The evolution of a partition of particles into clumps can be variously interpreted, for example, as a partition of atoms into molecules, a partition of molecules into polymers, or a partition of stars into galaxies. In a chemical setting, atoms coalesce to form molecules, and molecules coalesce to form polymers. In a cosmological setting, stars coalesce to form galaxies. The proposed research focuses on models for an irreversible process of coalescence of a partition of a continuum with finite total mass into a finite or countable collection of sub-masses, with conservation of mass.