Stochastic flocking dynamics of the inertial spin model with state-dependent noises

We study stochastic flocking dynamics of the inertial spin (IS) model with state-dependent noises. The IS model was considered to describe the collective behaviors of starling flocks moving with constant speed. Unlike mechanical flocking models extensively studied in the literature, this model incorporates an internal dynamic observable, namely spin (internal angular momentum) in addition to mechanical observables (position and velocity), and it describes how spin interacts with mechanical observables. In previous works, emergent dynamics of the deterministic counterparts for the IS model and its mean-field limit have been investigated under some specific setting in which network topology is multiplicatively separable. In this work, we present sufficient frameworks for stochastic flocking dynamics of the IS model, which state-dependent noises vanish at the equilibria of the deterministic IS model. The proposed frameworks are in terms of coupling strength, friction, and inertial coefficients, and our asymptotic convergence results for sample paths are given in both an almost sure and an expectation sense. We have also conducted several numerical experiments to verify our analytical results and to explore what can be studied further in future work.