A note on spectral gap and weighted Poincaré inequalities for some one-dimensional diffusions
Bonnefont, Michel1 ; Joulin, Aldéric2 ; Ma, Yutao3
1 Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351 cours de la Libération, 33405 Talence, France.
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.
ESAIM: Probability and Statistics, Tome 20 (2016), pp. 18-29.

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.

Reçu le :
DOI : 10.1051/ps/2015019
Classification : 60J60, 39B62, 37A30
Mots-clés : Spectral gap, diffusion operator, weighted Poincaré inequality
Bonnefont, Michel 1 ; Joulin, Aldéric 2 ; Ma, Yutao 3

1 Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351 cours de la Libération, 33405 Talence, France.
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. 
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     title = {A note on spectral gap and weighted {Poincar\'e} inequalities for some one-dimensional diffusions},
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     pages = {18--29},
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Bonnefont, Michel; Joulin, Aldéric; Ma, Yutao. A note on spectral gap and weighted Poincaré inequalities for some one-dimensional diffusions. ESAIM: Probability and Statistics, Tome 20 (2016), pp. 18-29. doi : 10.1051/ps/2015019. http://www.numdam.org/articles/10.1051/ps/2015019/

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