Vinh Nguyen, Roman Shvydkoy, Changhui Tan

Abstract

We study the mean-field limit for a class of agent-based models describing flocking with nonlinear velocity alignment. Each agent interacts through a communication protocol \(\phi\) and a non-linear coupling of velocities given by the power law \(A(v) = |v|^{p-2}v\), \(p>2\). The mean-field limit is proved in two settings -- deterministic and stochastic. We then provide quantitative estimates on propagation of chaos for deterministic case in the case of the classical fat-tailed kernels, showing an improved convergence rate of the \(k\)-particle marginals to a solution of the corresponding Vlasov equation. The stochastic version is addressed with multiplicative noise depending on the local interaction intensity, which leads to the associated Fokker-Planck-Alignment equation. Our results extend the classical Cucker-Smale theory to the nonlinear framework which has received considerable attention in the literature recently.

This work is supported by NSF grants DMS #2238219