# 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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