Logarithmic Sobolev inequalities for inhomogeneous Markov Semigroups

Jean-François Collet; Florent Malrieu

ESAIM: Probability and Statistics (2008)

  • Volume: 12, page 492-504
  • ISSN: 1292-8100

Abstract

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We investigate the dissipativity properties of a class of scalar second order parabolic partial differential equations with time-dependent coefficients. We provide explicit condition on the drift term which ensure that the relative entropy of one particular orbit with respect to some other one decreases to zero. The decay rate is obtained explicitly by the use of a Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's Γ-calculus. As a byproduct, the systematic method for constructing entropies which we propose here also yields the well-known intermediate asymptotics for the heat equation in a very quick way, and without having to rescale the original equation.

How to cite

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Collet, Jean-François, and Malrieu, Florent. "Logarithmic Sobolev inequalities for inhomogeneous Markov Semigroups." ESAIM: Probability and Statistics 12 (2008): 492-504. <http://eudml.org/doc/250396>.

@article{Collet2008,
abstract = { We investigate the dissipativity properties of a class of scalar second order parabolic partial differential equations with time-dependent coefficients. We provide explicit condition on the drift term which ensure that the relative entropy of one particular orbit with respect to some other one decreases to zero. The decay rate is obtained explicitly by the use of a Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's Γ-calculus. As a byproduct, the systematic method for constructing entropies which we propose here also yields the well-known intermediate asymptotics for the heat equation in a very quick way, and without having to rescale the original equation. },
author = {Collet, Jean-François, Malrieu, Florent},
journal = {ESAIM: Probability and Statistics},
keywords = {Inhomogeneous Markov process; logarithmic Sobolev inequality; relative entropy; inhomogeneous Markov process; logarithmic Sobolev inequality},
language = {eng},
month = {11},
pages = {492-504},
publisher = {EDP Sciences},
title = {Logarithmic Sobolev inequalities for inhomogeneous Markov Semigroups},
url = {http://eudml.org/doc/250396},
volume = {12},
year = {2008},
}

TY - JOUR
AU - Collet, Jean-François
AU - Malrieu, Florent
TI - Logarithmic Sobolev inequalities for inhomogeneous Markov Semigroups
JO - ESAIM: Probability and Statistics
DA - 2008/11//
PB - EDP Sciences
VL - 12
SP - 492
EP - 504
AB - We investigate the dissipativity properties of a class of scalar second order parabolic partial differential equations with time-dependent coefficients. We provide explicit condition on the drift term which ensure that the relative entropy of one particular orbit with respect to some other one decreases to zero. The decay rate is obtained explicitly by the use of a Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's Γ-calculus. As a byproduct, the systematic method for constructing entropies which we propose here also yields the well-known intermediate asymptotics for the heat equation in a very quick way, and without having to rescale the original equation.
LA - eng
KW - Inhomogeneous Markov process; logarithmic Sobolev inequality; relative entropy; inhomogeneous Markov process; logarithmic Sobolev inequality
UR - http://eudml.org/doc/250396
ER -

References

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