Fluctuations in Asymmetric Processes with Static and Dynamic Random Environments

9703811 Sethuraman 研究了在某些不对称随机动力系统中运动的单个或“标记”粒子的涨落行为。 随机系统分为两类模型：保守粒子系统和布朗捕获模型。 第一类包括简单的排除和零范围系统，而第二类则考虑“维也纳香肠”模型的变体。 这两个类别之间的一个重要区别在于，呈现给特定粒子的随机环境在第一个类别中“以可比较的规模变化”，而在第二个类别中“缓慢变化”。尽管关于这些模型的可逆版本，标记的粒子涨落已被很好地理解，但人们对这些过程何时具有漂移知之甚少。 该项目的重点是开发各种不依赖于对称性的估计和极限定理，从而识别不可逆的标记粒子涨落图。 这项工作解决的两个示例问题是：管理典型乘客与在复杂的海关队列中移动的其他乘客互动的行为的有效统计数据是什么？考虑一个洞穴的厚厚的多孔石头屋顶；雨滴穿过石头的裂缝滴入洞穴的机会有多大？采取什么策略才能避免被石头吸收？ 这两个问题符合所谓的“标记”粒子问题的框架，这是该项目的主要主题。 抽象地说，粒子（例如乘客或雨滴）与随机环境相互作用。 该项目的目标是了解各种不对称系统中粒子相互作用的波动概率。 这项工作的动机是，许多感兴趣的物理现象中最常见的情况，例如流体流动、排队和化学离子捕获行为，都可以通过这些不对称过程很好地建模。 该项目的意义在于，不对称性是所考虑模型的主要特征；以前，研究是在不太令人满意的对称性假设下进行的，这种假设常常简化模型而无法应用。

9703811 Sethuraman The fluctuation behavior of a single, or "tagged" particle moving in certain asymmetric stochastic dynamical systems is studied. The stochastic systems fall into two model classes: Conservative particle systems, and Brownian trapping models. The first category includes the simple exclusion and zero-range systems, while the second category considers variants of the "Wiener Sausage" model. An important difference between these two classes lies in that the random environments presented to a particular particle are "changing in a comparable scale" in the first and "slowly varying" in the second. Although the tagged particle fluctuations are well understood with respect to reversible versions of these models, little is known when these processes possess a drift. This project focuses on developing various estimates and limit theorems which do not depend on symmetry, so as to identify the non-reversible tagged particle fluctuation picture. Two sample questions which are addressed in this work are: What are the effective statistics governing the behavior of a typical passenger interacting with other passengers moving in a complex Customs queue? Consider a thick porous stone roof of a cave; what is the chance that a rain drop will weave its way through the fissures in the stone to drip into the cave, and what strategy does it take to avoid being absorbed into the stone? These two questions fit into the framework of the so-called "tagged" particle problem which is the main subject of this project. Abstractly, a particle, such as a passenger or a rain drop, interacts with a random environment. The goal of this project is to understand the fluctuation probabilities of this particle interaction in various systems with asymmetries. The motivation for this work is that the most general situations in many physical phenomena of interest, such as fluid flow, queuing, and chemical ion trapping behavior, are well modeled by these asymmetric p rocesses. The significance of this project is that asymmetry is a main feature of the models considered; previously, research proceeded under the less palatable assumption of symmetry which often simplified the models beyond application.