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

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.