Stochastic flocking dynamics of the Cucker–Smale model driven by Lévy noise

Lévy noise is a broader type of white noise, which describes discontinuous noise and is more closely to simulate the realistic environment. In this paper, we introduce Lévy noise into the Cucker–Smale model to investigate its effect on the flocking dynamics of the system. The well-posedness of the system is guaranteed by defining an appropriate stopping time and constructing a Lyapunov function. Through the strong law of large numbers for martingales, a sufficient framework of the strong stochastic flocking is obtained without the strict assumption that the communication rate function has a positive lower bound due to the introduction of Lévy noise. We find that Lévy noise can accelerate flocking and become good noise in some cases. Finally, several numerical simulations are presented to validate our results and further observe effects of Lévy noise.