We study functional inequalities (Poincar\'e, Cheeger, log-Sobolev) for probability measures obtained as perturbations. Several explicit results for general measures as well as log-concave distributions are given.The initial goal of this work was to obtain explicit bounds on the constants in view of statistical applications for instance. These results are then applied to the Langevin Monte-Carlo method used in statistics in order to compute Bayesian estimators.

翻译：我们研究了作为扰动的概率计量方法的功能不平等(Poincar\'e、Cheeger、log-Sobolev),给出了一些关于一般计量和日志集分布的明确结果。这项工作的最初目标是从统计应用等角度获得关于常数的明确界限。这些结果随后应用于统计中使用的Langevin Monte-Carlo方法,以便计算Bayesian估计数字。