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Semi Log-Concave Markov Diffusions

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 2123)

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 Γ 2 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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Cattiaux, P., Guillin, A. (2014). Semi Log-Concave Markov Diffusions. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLVI. Lecture Notes in Mathematics(), vol 2123. Springer, Cham. https://doi.org/10.1007/978-3-319-11970-0_9

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