Trends to equilibrium in total variation distance

Patrick Cattiaux; Arnaud Guillin

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 1, page 117-145
  • ISSN: 0246-0203

Abstract

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This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a given integrability condition. To this end, we give a general upper bound “à la Pinsker” enabling us to study our problem firstly via usual functional inequalities (Poincaré inequality, weak Poincaré,…) and truncation procedure, and secondly through the introduction of new functional inequalities ℐ ψ . These ℐ ψ -inequalities are characterized through measure-capacity conditions andF-Sobolev inequalities. A direct study of the decay of Hellinger distance is also proposed. Finally we show how a dynamic approach based on reversing the role of the semi-group and the invariant measure can lead to interesting bounds.

How to cite

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Cattiaux, Patrick, and Guillin, Arnaud. "Trends to equilibrium in total variation distance." Annales de l'I.H.P. Probabilités et statistiques 45.1 (2009): 117-145. <http://eudml.org/doc/78011>.

@article{Cattiaux2009,
abstract = {This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a given integrability condition. To this end, we give a general upper bound “à la Pinsker” enabling us to study our problem firstly via usual functional inequalities (Poincaré inequality, weak Poincaré,…) and truncation procedure, and secondly through the introduction of new functional inequalities $\mathcal \{I\}_\{\psi \}$. These $\mathcal \{I\}_\{\psi \}$-inequalities are characterized through measure-capacity conditions andF-Sobolev inequalities. A direct study of the decay of Hellinger distance is also proposed. Finally we show how a dynamic approach based on reversing the role of the semi-group and the invariant measure can lead to interesting bounds.},
author = {Cattiaux, Patrick, Guillin, Arnaud},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {total variation; diffusion processes; speed of convergence; Poincaré inequality; logarithmic Sobolev inequality; F-Sobolev inequality},
language = {eng},
number = {1},
pages = {117-145},
publisher = {Gauthier-Villars},
title = {Trends to equilibrium in total variation distance},
url = {http://eudml.org/doc/78011},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Cattiaux, Patrick
AU - Guillin, Arnaud
TI - Trends to equilibrium in total variation distance
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 1
SP - 117
EP - 145
AB - This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a given integrability condition. To this end, we give a general upper bound “à la Pinsker” enabling us to study our problem firstly via usual functional inequalities (Poincaré inequality, weak Poincaré,…) and truncation procedure, and secondly through the introduction of new functional inequalities $\mathcal {I}_{\psi }$. These $\mathcal {I}_{\psi }$-inequalities are characterized through measure-capacity conditions andF-Sobolev inequalities. A direct study of the decay of Hellinger distance is also proposed. Finally we show how a dynamic approach based on reversing the role of the semi-group and the invariant measure can lead to interesting bounds.
LA - eng
KW - total variation; diffusion processes; speed of convergence; Poincaré inequality; logarithmic Sobolev inequality; F-Sobolev inequality
UR - http://eudml.org/doc/78011
ER -

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