The Fokker-Planck equation with subcritical confinement force

Otared Kavian (1) , Stéphane Mischler , Mamadou Ndao (1)

1 LMV - Laboratoire de Mathématiques de Versailles

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 Röckner, 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.

Mots clés: Fokker-Planck equation; Krein-Rutman theorem; Semigroup; Spectral gap; Weak dissipativity; Weak Poincaré inequality

Domaines: Equations aux dérivées partielles [math.AP]

Otared Kavian, Stéphane Mischler, Mamadou Ndao. The Fokker-Planck equation with subcritical confinement force. Journal de Mathématiques Pures et Appliquées, 2021, 151, pp.171-211. ⟨10.1016/j.matpur.2021.04.007⟩. ⟨hal-03993027⟩