Séminaire de Probabilités XLVI pp 231-292

Part of the Lecture Notes in Mathematics book series (LNM, volume 2123) | Cite as

Semi Log-Concave Markov Diffusions

Chapter

Abstract

In this paper we intend to give a comprehensive approach of functional inequalities for diffusion processes under various “curvature” assumptions. One of them coincides with the usual Γ2 curvature of Bakry and Emery in the case of a (reversible) drifted Brownian motion, but differs for more general diffusion processes. Our approach using simple coupling arguments together with classical stochastic tools, allows us to obtain new results, to recover and to extend already known results, giving in many situations explicit (though non optimal) bounds. In particular, we show new results for gradient/semigroup commutation in the log concave case. Some new convergence to equilibrium in the granular media equation is also exhibited.

Keywords

Functional inequalities Transport inequalities Diffusion processes, Coupling Convergence to equilibrium 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institut de Mathématiques de ToulouseUniversité de Toulouse, CNRS UMR 5219Toulouse cedex 09France
  2. 2.Laboratoire de Mathématiques, CNRS UMR 6620Université Blaise PascalAubièreFrance

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