Semi Log-Concave Markov Diffusions

Patrick Cattiaux (1) , Arnaud Guillin (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LMBP - Laboratoire de Mathématiques Blaise Pascal

In this paper we intend to give a comprehensive approach of functional inequalities for diffusion processes under some ''curvature'' assumptions. Our notion of curvature coincides with the usual $\Gamma_2$ 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.

Mots clés: Functional inequalities; transport inequalities; diffusion processes; coupling; convergence to equilibrium

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Patrick Cattiaux, Arnaud Guillin. Semi Log-Concave Markov Diffusions. Séminaire de probabilités XLVI, 2123, Springer, pp 231-292, 2014, Lecture notes in mathematics. ⟨hal-00805299⟩