A note on spectral gap and weighted Poincar\'e inequalities for some one-dimensional diffusions

Michel Bonnefont (1) , Aldéric Joulin (2) , Yutao Ma (3)

1 IMB - Institut de Mathématiques de Bordeaux
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 BNU - Beijing Normal University

We present some classical and weighted Poincar\'e 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 Poincar\'e 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.

Domaines: Probabilités [math.PR]

Michel Bonnefont, Aldéric Joulin, Yutao Ma. A note on spectral gap and weighted Poincar\'e inequalities for some one-dimensional diffusions. ESAIM: Probability and Statistics, 2016, 20. ⟨hal-01082182⟩