THE KINETIC FOKKER-PLANCK EQUATION WITH MEAN FIELD INTERACTION

Arnaud Guillin (1) , Wei Liu (2) , Liming Wu (1) , Chaoen Zhang (1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 Wuhan University [China]

We study the long time behaviour of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fkker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H 1 (µ) with a rate of convergence which is explicitly computable and independent of the number of particles. The originality of the proof relies on functional inequalities and hypocoercivity with Lyapunov type conditions, usually not suitable to provide adimensional results.

Mots clés: Hypocoercivity; mean field interaction; Poincaré inequalities; logarithmic Sobolev inequality; Lyapunov conditions

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Arnaud Guillin, Wei Liu, Liming Wu, Chaoen Zhang. THE KINETIC FOKKER-PLANCK EQUATION WITH MEAN FIELD INTERACTION. Journal de Mathématiques Pures et Appliquées, 2021, 150, pp.1-23. ⟨10.1016/j.matpur.2021.04.001⟩. ⟨hal-02387517⟩