Logarithmic Sobolev inequalities for inhomogeneous Markov Semigroups
Laboratoire J.A. Dieudonné, Université de Nice Sophia
Antipolis, Parc Valrose, 06108 Nice Cedex 02, France.
2 IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France.
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.
Mathematics Subject Classification: 60J60 / 47D07
Key words: Inhomogeneous Markov process / logarithmic Sobolev inequality / relative entropy
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