Uniform Poincar{\'e} and logarithmic Sobolev inequalities for mean field particles systems

TL;DR

This paper derives explicit, uniform spectral gap and log-Sobolev inequalities for mean field particle systems with complex potentials, enabling precise exponential convergence rates for related equations.

Contribution

It provides sharp, explicit estimates for spectral gaps and log-Sobolev constants in mean field systems with multiple minima, extending existing convergence results.

Findings

Uniform spectral gap and log-Sobolev inequalities established

Exponential convergence in entropy with explicit rate derived

Applicable to systems with complex confinement potentials

Abstract

In this paper we establish some explicit and sharp estimates of the spectral gap and the log-Sobolev constant for mean field particles system, uniform in the number of particles, when the confinement potential have many local minimums. Our uniform log-Sobolev inequality, based on Zegarlinski's theorem for Gibbs measures, allows us to obtain the exponential convergence in entropy of the McKean-Vlasov equation with an explicit rate constant, generalizing the result of [10] by means of the displacement convexity approach, or [19, 20] by Bakry-Emery technique or the recent [9] by dissipation of the Wasserstein distance.