The Kinetic Fokker-planck Equation With Mean Field Interaction

Abstract

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 ($\mu$) 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.

References in corpus (5)

• Introduction to hypocoercive methods and applications for simple linear inhomogeneous kinetic models
• Hypocoercivity in Wasserstein-1 for the kinetic Fokker-Planck equation via Malliavin Calculus
• Ergodicity and propagation of chaos for mean field kinetic particles
• Gamma calculus beyond Villani and explicit convergence estimates for Langevin dynamics with singular potentials
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Cited by in corpus (4)

• Uniform long-time and propagation of chaos estimates for mean field kinetic particles in non-convex landscapes
• Convergence of unadjusted Hamiltonian Monte Carlo for mean-field models
• L2-Hypocoercivity and large time asymptotics of the linearized Vlasov-Poisson-Fokker-Planck system
• Ergodicity of the underdamped mean-field Langevin dynamics