  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.
