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

非线性马尔可夫过程、大型弱相互作用粒子系统及应用

• 批准号: 1305120
• 负责人: Amarjit Budhiraja
• 金额: $ 30万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1305120&HistoricalAwards=false
• 关键词: Nonlinear; Markov; processes; large; weakly

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.

本文考虑了具有马尔可夫动力学的大量弱相互作用粒子的系统，以及在大粒子极限下出现的非线性马尔可夫过程。这种系统最初是在统计力学中考虑的，然而近年来在许多不同领域的研究导致了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。基本的数学目标是对粒子集合的轨迹进行经验测量。针对这一问题，人们在大数行为定律（LLN）、中心极限理论和大偏差结果等方面做了大量的工作。例如，在某些条件下，在大粒子极限下的测量收敛于通过称为McKean-Vlasov方程的非线性演化方程表征的确定性测量。现有的大多数理论关注的是系统在有限时间范围内的行为。在提出的工作中，感兴趣的是经验测量过程的长时间行为及其LLN极限。更准确地说，目标是为相关的Mckean-Vlasov方程建立一个系统稳定性理论，并研究其对相互作用粒子系统的时间渐近行为的影响。将研究三种特定的模型族：(A)由通信网络产生的有限状态马尔可夫系统；(B)活性生物运输模型；(C)意见动态模型。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，这些协议可以使状态过程在长时间内保持在理想的运行条件附近。在来自生物系统的应用中，人们主要对描述系统稳定状态下的聚集、自组织和其他模式形成感兴趣。在社会科学应用中，如意见动态建模，关键目标之一是理解长期共识形成机制。所有这些主题都有一个共同的特点，即它们都与相关的经验测量过程的大时间限制的行为有关，这是本研究的中心目标。(A)的研究将导致改进复杂通信网络的设计、稳定性和管理的想法。(B)的研究将为在生物系统中观察到的各种模式形成提供见解和理解。(C)方面的研究将有助于制定最小干预方案，从而形成理想的长期共识模式。