Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Lipschitzian norm estimate of one-dimensional Poisson equations and applications

Hacene Djellout and Liming Wu

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Abstract

By direct calculus we identify explicitly the Lipschitzian norm of the solution of the Poisson equation $-\mathcal{L}G=g$ in terms of various norms of g, where $\mathcal{L}$ is a Sturm–Liouville operator or generator of a non-singular diffusion in an interval. This allows us to obtain the best constant in the L1-Poincaré inequality (a little stronger than the Cheeger isoperimetric inequality) and some sharp transportation–information inequalities and concentration inequalities for empirical means. We conclude with several illustrative examples.

Résumé

Par un calcul direct, on identifie explicitement la norme Lipschitzienne de la solution de l’équation de Poisson $-\mathcal {L}G=g$ en terme de différentes normes de g, où $\mathcal{L}$ est l’opérateur de Sturm–Liouville ou le générateur d’une diffusion non singulière sur un intervalle. Ainsi, nous pouvons obtenir, d’une part la meilleure constante dans l’inégalité de Poincaré L1 (une inégalité un peu plus forte que l’inégalité isopérimétrique de Cheeger) et d’autre part certaines inégalités de transport-information et de concentration fines pour la moyenne empirique. On conclut avec des exemples illustratifs.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 2 (2011), 450-465.

Dates
First available in Project Euclid: 23 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1300887277

Digital Object Identifier
doi:10.1214/10-AIHP360

Mathematical Reviews number (MathSciNet)
MR2814418

Zentralblatt MATH identifier
1233.47029

Subjects
Primary: 47B38: Operators on function spaces (general) 60E15: Inequalities; stochastic orderings 60J60: Diffusion processes [See also 58J65] 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds 35P15: Estimation of eigenvalues, upper and lower bounds

Keywords
Poisson equations Transportation–information inequalities Concentration and isoperimetric inequalities

Citation

Djellout, Hacene; Wu, Liming. Lipschitzian norm estimate of one-dimensional Poisson equations and applications. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 2, 450--465. doi:10.1214/10-AIHP360. https://projecteuclid.org/euclid.aihp/1300887277


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