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 -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.
Keywords: inhomogeneous Markov process, logarithmic Sobolev inequality, relative entropy
@article{PS_2008__12__492_0,
author = {Collet, Jean-Fran\c{c}ois and Malrieu, Florent},
title = {Logarithmic {Sobolev} inequalities for inhomogeneous {Markov} semigroups},
journal = {ESAIM: Probability and Statistics},
pages = {492--504},
year = {2008},
publisher = {EDP-Sciences},
volume = {12},
doi = {10.1051/ps:2007042},
mrnumber = {2455891},
zbl = {1187.60059},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/ps:2007042/}
}
TY - JOUR AU - Collet, Jean-François AU - Malrieu, Florent TI - Logarithmic Sobolev inequalities for inhomogeneous Markov semigroups JO - ESAIM: Probability and Statistics PY - 2008 SP - 492 EP - 504 VL - 12 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/ps:2007042/ DO - 10.1051/ps:2007042 LA - en ID - PS_2008__12__492_0 ER -
%0 Journal Article %A Collet, Jean-François %A Malrieu, Florent %T Logarithmic Sobolev inequalities for inhomogeneous Markov semigroups %J ESAIM: Probability and Statistics %D 2008 %P 492-504 %V 12 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/ps:2007042/ %R 10.1051/ps:2007042 %G en %F PS_2008__12__492_0
Collet, Jean-François; Malrieu, Florent. Logarithmic Sobolev inequalities for inhomogeneous Markov semigroups. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 492-504. doi: 10.1051/ps:2007042
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