The stochastic flocking model with far-field degenerate communication

Abstract

In this paper, we reconsider the stochastic kinetic Cucker-Smale model with multiplicative Brownian noise, in which we remove the positive lower bound assumption on the communication. In this case, the communication weight may vanish at the far field and the system is no longer controlled by a linear dissipative system. Then, we apply proper a priori estimates to prove the emergence of conditional flocking in strong sense, which naturally involves classical results for the deterministic model when noise intensity tends to zero. Moreover, we show that unconditional strong flocking occurs when communication weight decays slowly at the far field. Finally, these time asymptotical results imply uniform stability and mean-field limit. In particular, strong stability and mean-field limit in the expectation sense are established for the one-dimensional model via Lyapunov functional approach.