Flocking under Fast and Large Jumps: Stability, Chaos, and Traveling Waves

TL;DR

This paper analyzes a stochastic flocking model with large, fast jumps, characterizes its large system limit, and proves convergence to traveling wave solutions, addressing open problems in the field.

Contribution

It extends the understanding of flocking models to unbounded jump rates, establishing large system limits, stationary distributions, and convergence to traveling waves.

Findings

Large n limit of the empirical measure is characterized.

Existence and uniqueness of stationary distributions are proven.

Convergence to traveling wave solutions is established.

Abstract

We study a model for flocking given by a $n$-particle system under which each particle jumps forward by a random amount, independently sampled from a given distribution $\theta$, with rate given by a non-increasing function $w$ of its signed distance from the system center of mass. This model was introduced in Bal\'azs et. al. (2014) and some of its properties were studied for the case when $w$ is bounded. In the current work we are interested in the setting where $w$ is unbounded, and this feature results in a stochastic dynamical system for interacting particles with fast and large jumps for which little is available in the literature. We characterize the large $n$ limit (the so-called `fluid limit') of the empirical measure process associated with the system and prove a propagation of chaos result. Next, for the centered $n$-particle system, by constructing suitable Lyapunov functions, we establish existence and uniqueness of stationary distributions and study their tail properties. In the special case where $w$ is an exponential function and $\theta$ is an exponential distribution, by establishing that all stationary solutions of the McKean-Vlasov equation must be the unique fixed point of the equation, we prove a propagation of chaos result at $t=\infty$ and establish convergence of the particle system, starting from stationarity, in the large $n$ limit, to a traveling wave solution of the McKean-Vlasov equation. The proof of this result may be of interest for other interacting particle systems where convexity properties or functional inequalities generally used for establishing such a result are not available. Our work answers several open problems posed in Bal\'azs et. al.