Long Time Asymptotics for the Stochastic Follow-the-Leader System

TL;DR

This paper introduces a non-diffusive, pure-jump particle system that blends stochastic rat-race dynamics with a follow-the-leader structure on the real line. By focusing on the gap process, it proves a unique stationary distribution and uniform geometric ergodicity for finite particle systems, and, in the special case of Exp(1) leader jumps, derives an explicit product-form stationary law and sharp mixing-time bounds that scale as $t_{mix}=\Theta(n)$ lower and $O(n(\log n)^2)$ upper. The authors develop a novel coupling and a frozen-boundaries technique to analyze convergence and prove a functional central limit theorem under large-particle limits, with a moment-regularization phenomenon for heavy-tailed leader increments. Together, these results provide a tractable framework for ergodicity, invariant laws, and mixing behavior in local-interaction, non-diffusive particle systems and shed light on connections to Hammersley-type processes and potential hydrodynamic limits.

Abstract

We introduce and analyze a class of interacting particle systems on the real line that combine features of the stochastic rat race and (deterministic) follow-the-leader models. The particle system evolves as a continuous-time pure jump process: the leading particle moves independently, at Exponential jump times, with constant jump rate and iid jump sizes distributed according to a law $θ$, while each of the remaining particles jumps forward, at Exponential times, at rate equal to its distance from the particle immediately ahead, with jump sizes drawn uniformly from the corresponding gap. The dynamics thus encode competition for leadership together with distance-dependent stochastic interactions. Our main focus is the associated gap process, representing the vector of inter-particle distances. We establish the existence of a unique stationary distribution for the gap process and prove uniform geometric ergodicity. Further, when the leader's jump sizes follow an Exponential distribution, we identify the stationary law explicitly as a product of independent Exponential laws, and show that the associated mixing time scales between $Θ(n)$ and $O(n(\log n)^2)$ for an $n$-particle system. As an application of the mixing time results we establish a functional limit theorem that characterizes fluctuations of particle states at large time, under a suitable spatial and temporal scaling and large particle limit. Finally, when the leader's jumps have heavy but integrable tails, we show that each gap has at least one additional finite moment under stationarity than that of the leader's jump size distribution. The model offers a tractable setting for exploring ergodicity, explicit invariant laws, and mixing behavior in non-diffusive particle systems.

Theorems & Definitions (21)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Remark 3.4
• Theorem 3.5
• Corollary 3.6
• Theorem 3.7
• Lemma 5.1
• proof
• Lemma 5.2