Mean field systems on networks, with singular interaction through hitting times
Abstract
Building on the line of work [DIRT15a], [DIRT15b], [NS17a], [DT17], [HLS18], [HS18] we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very general driving processes and interaction functions, (ii) allow for inhomogeneous connection structures, and (iii) analyze a game in which the particles determine their connections strategically. Hereby, we uncover two completely new phenomena. First, we characterize the "times of fragility" of such systems (e.g., the times when a macroscopic part of the population defaults or gets infected simultaneously, or when the neuron cells "synchronize") explicitly in terms of the dynamics of the driving processes, the current distribution of the particles' values, and the topology of the underlying network (represented by its PerronFrobenius eigenvalue). Second, we use such systems to describe a dynamic creditnetwork game and show that, in equilibrium, the system regularizes: i.e., the times of fragility never occur, as the particles avoid them by adjusting their connections strategically. Two auxiliary mathematical results, useful in their own right, are uncovered during our investigation: a generalization of Schauder's fixedpoint theorem for the Skorokhod space with the M1 topology, and the application of the maxplus algebra to the equilibrium version of the network flow problem.
 Publication:

arXiv eprints
 Pub Date:
 July 2018
 arXiv:
 arXiv:1807.02015
 Bibcode:
 2018arXiv180702015N
 Keywords:

 Mathematics  Probability;
 Mathematics  Analysis of PDEs;
 Mathematics  General Topology;
 Quantitative Finance  Mathematical Finance;
 82C22;
 91A25;
 35K40;
 54H25;
 15A80
 EPrint:
 37 pages