Exponential relaxation to equilibrium for a kinetic Fokker-Planck-Alignment equation with force

Abstract

In this note, we consider a kinetic Fokker-Planck-Alignment equation with Rayleigh-type friction and self-propulsion force which is derived from general environmental averaging models. We show the exponential relaxation in time toward equilibrium of the solutions provided certain spectral gap conditions are satisfied. The result is proved by using Desvillettes-Villani's method for collisional models to establish the global hypocoercivity.