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

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 bas...

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Bibliographic Details
Published inProbability and statistics Vol. 20; pp. 18 - 29
Main Authors Bonnefont, Michel, Joulin, Aldéric, Ma, Yutao
Format Journal Article
LanguageEnglish
Published Les Ulis EDP Sciences 2016
Subjects
37A30
39B62
60J60
diffusion operator
Inequalities
Markov processes
Spectra
Spectral gap
weighted Poincaré inequality
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Summary: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.
Bibliography:MB is partially supported by the French ANR-12-BS01-0013-02 HAB project.
ark:/67375/80W-XXX64Q82-0
publisher-ID:ps150019
PII:S1292810015000191
istex:37F88CE44E0AD3BF3C48D2DAEE9DDFA83D3D14F6
ISSN:1292-8100
1262-3318
DOI:10.1051/ps/2015019