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Logarithmic Sobolev inequalities for inhomogeneous Markov Semigroups

Published online by Cambridge University Press:  01 November 2008

Jean-François Collet  and
Florent Malrieu
Jean-François Collet
Affiliation:
Laboratoire J.A. Dieudonné, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France.
Florent Malrieu
Affiliation:
IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France.
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Abstract

We investigate the dissipativity properties of a class of scalar secondorder parabolic partial differential equations with time-dependentcoefficients. We provide explicit condition on the drift term which ensurethat the relative entropy of one particular orbit with respect to some otherone decreases to zero. The decay rate is obtained explicitly by the use ofa 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 entropieswhich 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.

Keywords

Inhomogeneous Markov processlogarithmic Sobolev inequalityrelative entropy
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 492 - 504
Copyright
© EDP Sciences, SMAI, 2008

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