Brownian Motion and Models of Fragmentation and Coalescence

皮特曼和尤尔正在继续他们的工作，明确描述布朗运动和相关过程的各种功能的分布，如布朗和贝塞尔桥，弯曲和短途，并开发获得这种描述的新方法。布朗偏移和布朗桥在随机组合对象（如树和映射）的渐近性中的应用，与随机分区和随机离散分布理论的联系，以及布朗运动在数学金融中的应用，为获得更复杂的布朗泛函的精确分布和理解各种恒等提供了刺激。虽然现在已知了大量的显式公式，但仍然存在许多神秘的分布巧合，这些巧合在过去刺激了新技术的发展，并通过路径变换和分解等手段对其进行了更深入的理解。该奖项将继续研究随机过程，特别是布朗运动、随机分区和凝聚过程。布朗运动为具有连续路径的连续时间随机过程的现代理论提供了基础，并在物理和数学金融等各个领域都有应用。随机分区在组合学、物理学和遗传学中都有应用。聚结过程模拟随机现象，包括不可逆聚类或聚集在各种情况下。这是对随机过程数学结构的基础研究。这方面的进展增强了我们对这些过程的理解，并有可能应用于许多知识领域。

Pitman and Yor are continuing their work on explicit descriptions of the distribution of various functionals of Brownian motion and related processes such as Brownian and Bessel bridges, meanders and excursions, and the development of novel methods for obtaining such descriptions. Stimulus for obtaining the exact distributions of ever more complicated Brownian functionals, and understanding various identities, has been provided by applications of Brownian excursion and Brownian bridge to the asymptotics of random combinatorial objects such as trees and mappings, connections with the theory of random partitions and random discrete distributions, and the applications of Brownian motion in mathematical finance. While a great number of explicit formulae are now known there remain many mysterious distributional coincidences of the kind which have in the past provided stimulus for the development of novel techniques and deeper understanding through such devices as path transformations and decompositions.This award will continue present lines of research into stochastic processes, particularly Brownian motion, random partitions, and coalescent processes. Brownian motion provides the foundation 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. Coalescent processes model random phenomena involving irreversible clustering or aggregation in a wide variety of contexts. This is fundamental research into the mathematical structure of stochastic processes. Progress in this direction enhances our understanding of these processes, and has potential for application in numerous fields of knowledge.