Asymptotics for Particle Systems with Topological Interactions

根据该奖项开展的研究集中在随机动力系统上，该系统描述了一组相互作用的粒子的时间演化，其中最极端的粒子，例如最左边或最远的粒子，与系统中的其余粒子具有不同的动力学特征。这类系统产生于排队网络、进化生物学、数学金融学和其他科学领域的问题。这项工作的目标是了解当粒子数量变大和/或系统长时间运行时系统的行为。人们感兴趣的是描述典型的行为，典型行为的波动概率，以及非典型的与预期行为的大偏差。这项工作将导致对基于确定性模型的预测与实际噪声系统的偏差的理解，并为包含相关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洞察。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量的流体动力学极限可以用Stefan类自由边界问题的偏微分方程组(PDE)来描述。与这类粒子系统相关的几种类型的渐近问题将被研究。其中包括流体动力学极限、扩散近似、大偏差原理，以及在适当的空间、体积和时间尺度下的遍历行为。关于大偏差的工作将需要发展测值过程族的理论，其标度极限通过FBP来描述。一个关键的组成部分将是受控FBP族的唯一性理论的发展，以及与某些变分问题相关的Euler-Lagrange方程的分析。在另一个方向上，我们将研究这些系统的某些无限粒子形式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外，它也是在大型排队系统的负载平衡算法中的应用。无限粒子系统的多重不变分布的行为在其他相互作用的粒子系统中也是相同的。例子包括与某些类型粒子系统的标度极限有关的FBP行波解的多重性，以及与由Fleming-Viot型粒子系统近似的吸收的马尔可夫过程有关的拟平稳分布的多重性。这些相关的多样性现象将通过研究不同类型的初始配置下的比例限制以及时间和空间的比例来调查。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.