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

• 成果类型: Article
• 署名作者: Banerjee, Sayan; Budhiraja, Amarjit; Imon, Dilshad
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-026-01519-4
• 发表日期: 2026-08
• 页码: 921-995
• 关键词: Interacting particle systems; Attractive interaction; mean field; fluid limits; Topological interactions; flocking; long-time behavior; Traveling waves; Propagation of chaos; Interchange of limits; Asymptotic speed; particle-systems; propagation; CONVERGENCE; DIFFUSIONS

摘要

We study a pure-jump n-particle system with attractive mean field interactions under which each particle jumps forward by a random amount, independently sampled from a given distribution theta , at exponentially distributed times with rate given by a function w of its signed distance from the system center of mass. The function w is taken to be non-increasing which leads to a 'flocking' behavior: the particles below the center of mass jump forward at a higher rate than those above it. This model was introduced in [] 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, together with the mild integrability we impose on the jump sizes, results in a stochastic dynamical system for interacting particles with fast and large jumps for which little is available in the literature. We identify natural conditions under which the system is well-posed and study the large n limit (the so-called 'fluid limit') of the empirical measure process associated with the system. By establishing well-posedness of the associated McKean-Vlasov equation we characterize the fluid limit of the particle 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=infinity 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 [].