Quand est-ce que des bornes de Hardy permettent de calculer une constante de Poincaré exacte sur la droite ?

Laurent Miclo[1]

  • [1] Laboratoire d’Analyse, Topologie, Probabilités, UMR 6632, Centre de Mathématiques et Informatique, Université de Provence et Centre National de la Recherche Scientifique, 39, rue F. Joliot-Curie, 13453 Marseille cedex 13, France

Annales de la faculté des sciences de Toulouse Mathématiques (2008)

  • Volume: 17, Issue: 1, page 121-192
  • ISSN: 0240-2963

Abstract

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Classically, Hardy’s inequality enables to estimate the spectral gap of a one-dimensional diffusion up to a factor belonging to [ 1 , 4 ] . The goal of this paper is to better understand the latter factor, at least in a symmetric setting. In particular, we will give an asymptotical criterion implying that its value is exactly 4. The underlying argument is based on a semi-explicit functional for the spectral gap, which is monotone in some rearrangement of the data. To find it will resort to some regularity properties of Poincaré’s constant and we will exhibit some links, more or less already known, with path methods, principal Dirichlet eigenvalues, Sturm-Liouville’s equations and Brownian functionals. Finally, we will extend the investigation to the case of birth and death processes on ℤ , still in a symmetric context. We hope this approach can be extended to more difficult functional inequalities.

How to cite

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Miclo, Laurent. "Quand est-ce que des bornes de Hardy permettent de calculer une constante de Poincaré exacte sur la droite ?." Annales de la faculté des sciences de Toulouse Mathématiques 17.1 (2008): 121-192. <http://eudml.org/doc/10075>.

@article{Miclo2008,
abstract = {Classically, Hardy’s inequality enables to estimate the spectral gap of a one-dimensional diffusion up to a factor belonging to $[1,4]$. The goal of this paper is to better understand the latter factor, at least in a symmetric setting. In particular, we will give an asymptotical criterion implying that its value is exactly 4. The underlying argument is based on a semi-explicit functional for the spectral gap, which is monotone in some rearrangement of the data. To find it will resort to some regularity properties of Poincaré’s constant and we will exhibit some links, more or less already known, with path methods, principal Dirichlet eigenvalues, Sturm-Liouville’s equations and Brownian functionals. Finally, we will extend the investigation to the case of birth and death processes on $\mathbb\{Z\}$, still in a symmetric context. We hope this approach can be extended to more difficult functional inequalities.},
affiliation = {Laboratoire d’Analyse, Topologie, Probabilités, UMR 6632, Centre de Mathématiques et Informatique, Université de Provence et Centre National de la Recherche Scientifique, 39, rue F. Joliot-Curie, 13453 Marseille cedex 13, France},
author = {Miclo, Laurent},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Poincaré inequality; Hardy's inequality; path methods; principal Dirichlet eigenvalue; Sturm-Liouville equation; Brownian functionals},
language = {eng},
month = {6},
number = {1},
pages = {121-192},
publisher = {Université Paul Sabatier, Toulouse},
title = {Quand est-ce que des bornes de Hardy permettent de calculer une constante de Poincaré exacte sur la droite ?},
url = {http://eudml.org/doc/10075},
volume = {17},
year = {2008},
}

TY - JOUR
AU - Miclo, Laurent
TI - Quand est-ce que des bornes de Hardy permettent de calculer une constante de Poincaré exacte sur la droite ?
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2008/6//
PB - Université Paul Sabatier, Toulouse
VL - 17
IS - 1
SP - 121
EP - 192
AB - Classically, Hardy’s inequality enables to estimate the spectral gap of a one-dimensional diffusion up to a factor belonging to $[1,4]$. The goal of this paper is to better understand the latter factor, at least in a symmetric setting. In particular, we will give an asymptotical criterion implying that its value is exactly 4. The underlying argument is based on a semi-explicit functional for the spectral gap, which is monotone in some rearrangement of the data. To find it will resort to some regularity properties of Poincaré’s constant and we will exhibit some links, more or less already known, with path methods, principal Dirichlet eigenvalues, Sturm-Liouville’s equations and Brownian functionals. Finally, we will extend the investigation to the case of birth and death processes on $\mathbb{Z}$, still in a symmetric context. We hope this approach can be extended to more difficult functional inequalities.
LA - eng
KW - Poincaré inequality; Hardy's inequality; path methods; principal Dirichlet eigenvalue; Sturm-Liouville equation; Brownian functionals
UR - http://eudml.org/doc/10075
ER -

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