Download PDFOpen PDF in browserOn the Exponential Ergodicity of the McKeanVlasov SDE Depending on a Polynomial InteractionEasyChair Preprint no. 1032138 pages•Date: June 1, 2023AbstractIn this paper, we study the long time behaviour of the FokkerPlanck and the kinetic FokkerPlanck equations with many body interaction, more precisely with interaction defined by Ustatistics, whose macroscopic limits are often called McKeanVlasov and VlasovFokkerPlanck equations respectively. In the continuity of the recent papers [63, [43],[42]] and [44, [74],[75]], we establish nonlinear functional inequalities for the limiting McKeanVlasov SDEs related to our particle systems. In the first order case, our results rely on large deviations for Ustatistics 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 VlasovFokkerPlanck equation. Keyphrases: (hypo)coercivity, (kinetic) FokkerPlanck equation, 26D10, 39B62, 47D07, 60G10, 60H10, 60J60, 82C31, convergence to equilibrium, Functional inequalities, McKeanVlasov equation, polynomial interaction, propagation of chaos, Ustatistics
