On Stochastic Cucker-Smale flocking dynamics

TL;DR

This paper introduces a stochastic extension of the Cucker-Smale flocking model, analyzing its behavior as the number of particles grows large and establishing the uniqueness and propagation of chaos for the resulting stochastic equations.

Contribution

It develops a stochastic version of the Cucker-Smale model with unbounded interactions, proves the mean-field limit, and establishes uniqueness and propagation of chaos.

Findings

Established the infinite particle limit for the stochastic model.

Proved uniqueness in law for the Vlasov-McKean equation.

Quantified convergence using total variation and Wasserstein distances.

Abstract

We present a stochastic version of the Cucker-Smale flocking dynamics based on a markovian $N$-particle system of pair interactions with unbounded and, in general, non-Lipschitz continuous interaction potential. We establish the infinite particle limit $N \to \infty$ and identify the limit as a solution with a nonlinear martingale problem describing the law of a weak solution to a Vlasov-McKean stochastic equation with jumps. Moreover, we estimate the total variation and Wasserstein distance for the time-marginals from which uniqueness in the class of solutions having some finite exponential moments is deduced. Based on the uniqueness for the time-marginals we prove uniqueness in law for the Vlasov-McKean equation, i.e. we establish propagation of chaos.