paper

On the exponential ergodicity of the McKean-Vlasov SDE depending on a polynomial interaction

arXiv:2306.01347 · doi:10.1016/j.spa.2026.105088

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 [63, [43],[42]] and [44, [74],[75]], 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 H 1 () 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 W 2 --metric for the Vlasov-Fokker-Planck equation.

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