Mean Field Systems on Networks, with Singular Interaction Through Hitting Times

In this talk, I will present the recent advances in the study of particle systems with singular interaction through hitting times. Such systems appear naturally in the mathematical models for neuron cells, supercooled liquids, credit networks, etc. In contrast to the previous research, this work (i) allows for inhomogeneous connection structures, and (ii) constructs an equilibrium in 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 process, the current distribution of the particles' values, and the topology of the underlying network (represented by its Perron-Frobenius eigenvalue). Second, we use such systems to describe a dynamic credit-network 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 fixed-point theorem for the Skorokhod space with M1 topology, and the application of max-plus algebra to the equilibrium problem for credit networks.

Event Topic

Mathematical Finance, Stochastic Analysis, and Machine Learning