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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 Balázs 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 Balázs et. al.

Submission history

From: Sayan Banerjee [view email]
[v1] Fri, 19 Apr 2024 18:07:13 UTC (64 KB)
[v2] Mon, 13 Jul 2026 12:22:51 UTC (791 KB)