Nonlinear Markov processes, large weakly interacting particle systems, and applications

这项工作考虑了大量的弱相互作用的粒子与马尔可夫动力学系统，和非线性马尔可夫过程中出现的大粒子的限制。 这种系统最初被认为是在统计力学，但近年来在许多不同领域的研究已经导致类似的随机动力学模型。一些例子包括，损失网络模型，大型投资组合中的默认聚类，趋化反应动力学和社会科学中的信念系统。基本的数学对象是粒子集合轨迹的经验测量。对于该对象的大数行为定律（LLN）、中心极限理论和大偏差结果进行了大量工作。例如，在条件下，在大粒子极限下的这个测量收敛到确定性测量，其特征在于通过称为McKean-Vlasov方程的非线性演化方程。大多数现有的理论关注的是系统在有限时间范围内的行为。在拟议的工作中的兴趣是在长时间的行为的经验测量过程和它的LLN限制。更确切地说，目标是为相关的Mckean-Vlasov方程建立一个系统的稳定性理论，并研究其对相互作用粒子系统的时间渐近行为的影响。三个具体的家庭的模型将进行研究：（A）有限状态马尔可夫系统产生的通信网络;（B）模型的主动生物运输;（C）意见动力学模型。在许多应用中，上述形式的相互作用粒子系统的时间渐近行为是主要关注的问题。例如，在通信系统中，稳定性和控制是基本的，并且人们对在长时间段内将状态过程保持在期望的操作条件的邻域中的系统设计和控制协议感兴趣。在来自生物系统的应用中，人们主要感兴趣的是描述聚集，自组织和系统稳态中的其他模式形成。在社会科学应用中，如意见动态建模，关键目标之一是了解长期共识形成机制。所有这些主题都有一个共同的特点，即它们都与关联的经验测量过程的大时限行为有关，对这一问题的研究是本研究的中心目标。 （A）中的研究将导致复杂通信网络的改进设计、稳定性和监管的想法。（B）的研究将提供对生物系统中观察到的不同模式形成的洞察和理解。（C）中的研究将能够开发最小干预协议，从而导致理想的长期共识模式。

This work considers systems with a large number of weakly interacting particles with Markovian dynamics, and nonlinear Markov processes that arise in the large particle limit. Such systems originally were considered in statistical mechanics, however in recent years studies in many different fields have led to similar stochastic dynamical models. Some examples include, loss network models, default clustering in large portfolios, chemotactic response dynamics, and belief systems in social sciences. The basic mathematical object is the empirical measure of the trajectories of the collection of particles. There is extensive work on the law of large number behavior(LLN), central limit theory and large deviation results for this object. For example, under conditions, this measure under a large particle limit converges to a deterministic measure that is characterized through a nonlinear evolution equation known as the McKean-Vlasov equation. Most of the existing theory concerns the behavior of the system on a finite time horizon. In the proposed work the interest is in the long time behavior of the empirical measure process and its LLN limit. More precisely, the goal is to develop a systematic stability theory for the associated Mckean-Vlasov equation, and to study its consequences for the time asymptotic behavior of the interacting particle system. Three specific families of models will be studied: (A) Finite state Markovian systems arising from communication networks; (B) Models for active biological transport; (C) Opinion dynamics models. In many applications the time asymptotic behavior of an interacting particle system of the above form is of central concern. For example, in communication systems stability and control is fundamental and one is interested in system design and control protocols that keep the state processes in the neighborhoods of desirable operating conditions over long periods of time. In applications coming from biological systems, one is primarily interested in describing aggregation, self organization and other pattern formations in the steady state of the system. In social science applications, such as opinion dynamics modeling, one of the key goals is to understand long term consensus formation mechanisms. All of these topics have in common the feature that they are related to the behavior of the large time limit of the associated empirical measure process, the study of which is the central goal of this research. Research in (A) will lead to ideas for improved design, stability, and regulation of complex communication networks. Research in (B) will provide insight and understanding for diverse pattern formations observed in biological systems. Research in (C) will enable development of minimal intervention protocols that lead to desirable long term consensus patterns.