UNIFORM POINCARE AND LOGARITHMIC SOBOLEV INEQUALITIES FOR MEAN FIELD PARTICLE SYSTEMS

• 成果类型: Article
• 署名作者: Guillin, Arnaud; Liu, Wei; Wu, Liming; Zhang, Chaoen
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Clermont Auvergne (UCA); Wuhan University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1707
• 发表日期: 2022
• 页码: 1590-1614
• 关键词: unbounded spin systems; granular media equations; Stochastic dynamics; Simple proof; CONVERGENCE; equilibrium; EQUIVALENCE; rates

摘要

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 Poincare inequality is based on the work of Ledoux (In Seminaire de Probabilites, XXXV (2001) 167194, 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 McKeanVlasov 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) 109132; 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).