The Fokker-Planck equation with subcritical confinement force

Abstract: We consider the Fokker-Planck equation with subcritical confinement force field which may not derive from a potential function. We prove the existence of an equilibrium (in the case of a general force) and we establish some (polynomial and stretch exponential) rate of convergence to the equilibrium (depending on the space to which belongs the initial datum). Our results improve similar results established by Toscani, Villani [29] and R{\"o}chner , Wang [27]: the force field is more general, the spaces are more general, the rates are sharper.