The kinetic Fokker-Planck equation with mean field interaction

We study the long time behavior of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fokker-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(mu) 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. (C) 2021 Elsevier Masson SAS. All rights reserved.

Automated summary

The study proves a uniform exponential convergence to equilibrium for the solutions of the kinetic Fokker-Planck equation with mean field interaction in the weighted Sobolev space, with a computable rate of convergence independent of the number of particles. The originality of the proof lies in the use of functional inequalities and hypocoercivity with Lyapunov type conditions, typically not suitable for providing dimensionless results.