Weak logarithmic Sobolev inequalities and entropic convergence

• 成果类型: Article
• 署名作者: Cattiaux, P.; Gentil, I.; Guillin, A.
• 署名单位: Institut Polytechnique de Paris; Ecole Polytechnique; Universite Paris Nanterre; Universite PSL; Universite Paris-Dauphine; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Aix-Marseille Universite; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-007-0054-5
• 发表日期: 2007
• 页码: 563-603
• 关键词: functional inequalities; poincare; rates

摘要

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincare inequalities, general Beckner inequalities, etc.). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincare inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.