Abstract
In this paper we consider a mean field particle systems whose confinement potentials have many local minima. We establish some explicit and sharp estimates of the spectral gap and logarithmic Sobolev constants uniform in the number of particles. The uniform Poincaré inequality is based on the work of Ledoux (In Séminaire de Probabilités, XXXV (2001) 167–194, Springer) and the uniform logarithmic Sobolev inequality is based on Zegarlinski’s theorem for Gibbs measures, both combined with an explicit estimate of the Lipschitz norm of the Poisson operator for a single particle from (J. Funct. Anal. 257 (2009) 4015–4033). The logarithmic Sobolev inequality then implies the exponential convergence in entropy of the McKean–Vlasov equation with an explicit rate, We need here weaker conditions than the results of (Rev. Mat. Iberoam. 19 (2003) 971–1018) (by means of the displacement convexity approach), (Stochastic Process. Appl. 95 (2001) 109–132; Ann. Appl. Probab. 13 (2003) 540–560) (by Bakry–Emery’s technique) or the recent work (Arch. Ration. Mech. Anal. 208 (2013) 429–445) (by dissipation of the Wasserstein distance).
Funding Statement
W. Liu is supported by the NSFC 12071361 and 11731009, the Fundamental Research Funds for the Central Universities 2042020kf0031 and 2042020kf0217, and CSC.
Acknowledgements
Part of these results were first presented in the “Workshop on stability of functional inequalities and applications” in 2018 in Toulouse which is supported by the Labex CIMI and the ANR project “Entropies, Flots, Inegalites”. We sincerely thank the two referees and the Associate Editor for well-pointed comments which lead to a substantial improvement of the paper.
Citation
Arnaud Guillin. Wei Liu. Liming Wu. Chaoen Zhang. "Uniform Poincaré and logarithmic Sobolev inequalities for mean field particle systems." Ann. Appl. Probab. 32 (3) 1590 - 1614, June 2022. https://doi.org/10.1214/21-AAP1707
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