On the Exponential Ergodicity of the McKean-Vlasov SDE Depending on a Polynomial Interaction

EasyChair Preprint 10321, version 2

Abstract

In this paper, we study the long time behaviour of the Fokker-Planck and the kinetic Fokker-Planck equations with many body interaction, more precisely with interaction defined by U-statistics, whose macroscopic limits are often called McKean-Vlasov and Vlasov-Fokker-Planck equations respectively. In the continuity of the recent papers [65, 45, 44] and [46, 76, 77], we establish nonlinear functional inequalities for the limiting McKean-Vlasov SDEs related to our particle systems. In the first order case, our results rely on large deviations for U-statistics and a uniform logarithmic Sobolev inequality in the number of particles for the invariant measure of the particle system. In the kinetic case, we first prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H1(μ) with a rate of convergence which is explicitly computable and independent of the number of particles. In a second time, we quantitatively establish an exponential return to equilibrium in Wasserstein’s W2-metric for the Vlasov-Fokker-Planck equation. Some concrete examples are also provided.

Keyphrases: (hypo)coercivity, (kinetic) Fokker-Planck equation, 26D10, 39B62, 47D07, 60G10, 60H10, 60J60, 82C31, Functional inequalities, McKean-Vlasov equation, U-statistics, convergence to equilibrium, polynomial interaction, propagation of chaos

@booklet{EasyChair:10321,
  author    = {Mohamed Alfaki Ag Aboubacrine Assadeck},
  title     = {On the Exponential Ergodicity of the McKean-Vlasov SDE Depending on a Polynomial Interaction},
  howpublished = {EasyChair Preprint 10321},
  year      = {EasyChair, 2024}}