# Logarithmic Sobolev Inequalities for Inhomogeneous Markov Semigroups

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 $\Gamma-$ 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.
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https://hal.archives-ouvertes.fr/hal-00019623
Contributor : Florent Malrieu <>
Submitted on : Friday, February 24, 2006 - 11:42:53 AM
Last modification on : Friday, November 16, 2018 - 1:31:21 AM
Long-term archiving on: Saturday, April 3, 2010 - 10:34:20 PM

### Citation

Jean-François Collet, Florent Malrieu. Logarithmic Sobolev Inequalities for Inhomogeneous Markov Semigroups. ESAIM: Probability and Statistics, EDP Sciences, 2008, 12, pp.492-504. ⟨10.1051/ps:2007042⟩. ⟨hal-00019623⟩

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