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Asymptotics for Particle Systems with Topological Interactions

具有拓扑相互作用的粒子系统的渐近

基本信息

批准号：
2152577
负责人：
Amarjit Budhiraja
金额：
$ 32.99万
依托单位：
University of North Carolina at Chapel Hill
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2152577&HistoricalAwards=false
关键词：
Asymptotics Particle Systems Topological Interactions

项目摘要

Research carried out under this award centers around stochastic dynamical systems that describe the time evolution of a collection of interacting particles in which the most extreme particle, e.g. the leftmost or the farthest particle, has different dynamical signatures than the remaining particles in the system. Such systems arise from problems in queuing networks, evolutionary biology, mathematical finance, and other areas of science. The goal of the work is to understand the behavior of such systems as the number of particles becomes large and/or when the system is in operation for a long time. One is interested in characterizing typical behavior, probabilities of fluctuations from the typical behavior, and also of non-typical large deviations from the expected behavior. The work will lead to an understanding of divergence of predictions based on deterministic models from the actual noisy systems and provide guidance for operating procedures that incorporate the associated uncertainties. It will also provide insight on properties of these systems over suitable spatial and temporal scales. The project includes research training opportunities for graduate students. Under suitable scaling, hydrodynamical limits of empirical measures of such particle systems can be characterized through partial differential equations (PDE) for Stefan type free boundary problems (FBP). Several types of asymptotic problems associated with such particle systems will be studied. These include hydrodynamic limits, diffusion approximations, large deviation principles, and ergodicity behavior under suitable scaling of space, volume and time. Work on large deviations will require the development of the theory for families of measure valued processes whose scaling limits are described through FBP. A key component will be the development of the uniqueness theory for families of controlled FBP and an analysis of Euler-Lagrange equations associated with certain calculus of variations problems. In another direction, the local stability structure of extremal invariant distributions of some infinite particle versions of these systems will be studied. In addition to this being a fundamental problem in the ergodic theory of infinite dimensional Markov processes, here it also arises from applications to load balancing algorithms for large queuing systems. The behavior of multiplicity of invariant distributions for infinite particle systems has counterparts in other interacting particle systems. Examples include multiplicity of traveling wave solutions of FBP associated with scaling limits of certain types of particle systems, and multiplicity of quasi-stationary distributions associated with Markov processes with absorption approximated by Fleming-Viot type particle systems. These related multiplicity phenomena will be investigated by studying scaling limits under different types of initial configurations and time and space scaling.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该奖项下进行的研究围绕随机动力学系统，描述了一组相互作用粒子的时间演化，其中最极端的粒子，例如最左边或最远的粒子，具有与系统中其余粒子不同的动力学特征。这样的系统源于排队网络、进化生物学、数理金融和其他科学领域的问题。这项工作的目标是了解这种系统的行为，因为粒子的数量变得很大，和/或当系统运行了很长一段时间。人们感兴趣的是表征典型的行为，从典型的行为波动的概率，以及非典型的大偏差从预期的行为。这项工作将导致从实际的噪声系统的确定性模型的基础上的预测的分歧的理解，并结合相关的不确定性的操作程序提供指导。它还将提供对这些系统在适当的空间和时间尺度上的特性的见解。该项目包括为研究生提供研究培训机会。在适当的尺度下，这类粒子系统的经验测度的流体动力学极限可以通过Stefan型自由边界问题（FBP）的偏微分方程（PDE）来表征。与这种粒子系统相关的几种类型的渐近问题将被研究。这些包括流体动力学限制，扩散近似，大偏差原则，和遍历性行为下适当的缩放空间，体积和时间。大偏差的工作将需要发展的理论，家庭的措施价值的过程，其标度限制通过FBP描述。一个关键的组成部分将是家庭的控制FBP和欧拉-拉格朗日方程的分析与某些变分问题的唯一性理论的发展。在另一个方向上，将研究这些系统的一些无限粒子版本的极值不变分布的局部稳定性结构。除了这是无限维马尔可夫过程遍历理论中的一个基本问题之外，它还源于大型排队系统负载平衡算法的应用。无限粒子系统不变分布的多重性在其他相互作用粒子系统中也有类似的表现。例子包括与某些类型的粒子系统的标度极限相关联的FBP的行波解的多重性，以及与具有近似Fleming-Viot型粒子系统的吸收的马尔可夫过程相关联的准平稳分布的多重性。这些相关的多重性现象将通过研究不同类型的初始配置和时间和空间scaling.This奖项反映了NSF的法定使命的缩放限制进行调查，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。