Volume 20, 2016
|Page(s)||18 - 29|
|Published online||03 June 2016|
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
2 Université de Toulouse, Institut National des Sciences Appliquées, Institut de Mathématiques de Toulouse, 31077 Toulouse, France.
3 School of Mathematical Sciences & Lab. Math. Com. Sys., Beijing Normal University, 100875 Beijing, P.R. China.
Revised: 1 December 2015
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.
Mathematics Subject Classification: 60J60 / 39B62 / 37A30
Key words: Spectral gap / diffusion operator / weighted Poincaré inequality
© EDP Sciences, SMAI 2016
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