Uniform Poincaré and logarithmic Sobolev inequalities for mean field particles systems

2019/09/16 by Arnaud Guillin, Wei Liu, Guillin, Arnaud +5 · 1 citation
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Gas Dynamics and Kinetic Theory #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.1909.07051

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.

Citations

• Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

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• Exponential Ergodicity for McKean-Vlasov SDEs with Singular Interactions

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