The Fokker-Planck equation with subcritical confinement force

We consider the Fokker-Planck equation with subcritical confinement force field which may not derive from a potential function. We prove the existence of a unique positive equilibrium of mass one and we establish some subgeometric, or geometric, rate of convergence to a multiple of this equilibrium (depending on the space to which belongs the initial datum) in many spaces. Our results generalize similar results introduced by Toscani, Villani [33] and Riickner, Wang [31] for some forces associated to a potential and extended by Douc, Fort, Guillin [12] and Bakry, Cattiaux, Guillin [4] for some general forces: however in our approach the spaces are more general, and the rates of convergence to equilibrium are sharper. (C) 2021 Elsevier Masson SAS. All rights reserved.