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
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topCollet, 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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