Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

Abstract

We use probabilistic methods to study properties of mean-field models, arising as large-scale limits of certain particle systems with mean-field interaction. The underlying particle system is such that $n$ particles move forward on the real line. Specifically, each particle "jumps forward" at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle's location quantile within the overall distribution of particle locations. A mean-field model describes the evolution of the particles' distribution, when $n$ is large. It is essentially a solution to an integro-differential equation within a certain class. Our main results concern the existence and uniqueness of -- and attraction to -- mean-field models which are traveling waves, under general conditions on the jump-rate function and the jump-size distribution.