A note on spectral gap and weighted Poincaré inequalities for some one-dimensional diffusions

¹ Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351 cours de la Libération, 33405 Talence, France.
michel.bonnefont@math.u-bordeaux.fr
² Université de Toulouse, Institut National des Sciences Appliquées, Institut de Mathématiques de Toulouse, 31077 Toulouse, France.
ajoulin@insa-toulouse.fr
³ School of Mathematical Sciences & Lab. Math. Com. Sys., Beijing Normal University, 100875 Beijing, P.R. China.
mayt@bnu.edu.cn

Received: 16 January 2015
Revised: 1 December 2015

Abstract

We present some classical and weighted Poincaré inequalities for some one-dimensional probability measures. This work is the one-dimensional counterpart of a recent study achieved by the authors for a class of spherically symmetric probability measures in dimension larger than 2. Our strategy is based on two main ingredients: on the one hand, the optimal constant in the desired weighted Poincaré inequality has to be rewritten as the spectral gap of a convenient Markovian diffusion operator, and on the other hand we use a recent result given by the two first authors, which allows to estimate precisely this spectral gap. In particular we are able to capture its exact value for some examples.