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

这项工作考虑了具有大量弱相互作用粒子的马尔可夫动力学系统，以及在大粒子极限下出现的非线性马尔可夫过程。这类系统最初在统计力学中被考虑，但近年来在许多不同领域的研究导致了相似的随机动力学模型。一些例子包括，损失网络模型，大型投资组合中的违约聚集，趋化反应动力学，以及社会科学中的信念系统。基本的数学对象是对粒子集合的轨迹的经验测量。关于这个对象的大数行为定律(LLN)、中心极限理论和大偏差结果已经有了广泛的工作。例如，在一定条件下，大粒子极限下的这种度量收敛于通过称为McKean-Vlasov方程的非线性发展方程来表征的确定性度量。现有的大多数理论都是关于系统在有限时间范围内的行为。在所提出的工作中，感兴趣的是经验测量过程的长期行为及其Lln极限。更准确地说，我们的目标是发展一个关联的McKean-Vlasov方程的系统稳定性理论，并研究它对相互作用粒子系统的时间渐近行为的影响。将研究三类具体的模型：(A)由通信网络产生的有限状态马尔可夫系统；(B)主动生物运输模型；(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人们关注的中心问题。例如，在通信系统中，稳定性和控制是基本的，人们感兴趣的是系统设计和控制协议，这些协议使状态过程在较长时间内保持在期望的操作条件的邻域中。在生物系统的应用中，人们主要感兴趣的是描述系统稳定状态下的聚集、自组织和其他模式的形成。在社会科学应用中，如观点动力学建模，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，对此的研究是本研究的中心目标。在(A)中的研究将导致改进复杂通信网络的设计、稳定性和监管的想法。(B)中的研究将提供对生物系统中观察到的各种模式形成的洞察和理解。(C)中的研究将使最小干预方案的制定成为可能，从而产生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.