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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>, at exponentially distributed times with rate given by a function w of its signed distance from the system center of mass. The function w is taken to be non-increasing which leads to a ‘flocking’ behavior: the particles below the center of mass jump forward at a higher rate than those above it. This model was introduced in [<CitationRef CitationID="CR3">3</CitationRef>] and some of its properties were studied for the case when w is bounded. In the current work we are interested in the setting where w is unbounded, and this feature, together with the mild integrability we impose on the jump sizes, results in a stochastic dynamical system for interacting particles with fast and large jumps for which little is available in the literature. We identify natural conditions under which the system is well-posed and study the large n limit (the so-called ‘fluid limit’) of the empirical measure process associated with the system. By establishing well-posedness of the associated McKean-Vlasov equation we characterize the fluid limit of the particle system and prove a propagation of chaos result. Next, for the centered n-particle system, by constructing suitable Lyapunov functions, we establish existence and uniqueness of stationary distributions and study their tail properties. In the special case where w is an exponential function and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation> is an exponential distribution, by establishing that all stationary solutions of the McKean-Vlasov equation must be the unique fixed point of the equation, we prove a propagation of chaos result at <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$t=\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> 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 [<CitationRef CitationID="CR3">3</CitationRef>].

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

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发表日期：

2026年7月18日

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Sayan Banerjee,
Amarjit Budhiraja,
Dilshad Imon

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 $ $θ$ , 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 [3] 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 $ $θ$ 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 $ $t = \infty$ 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 [3].