Transportation-information inequalities for Markov processes (II) : relations with other functional inequalities

Arnaud Guillin (1) , Christian Léonard (2, 3) , Feng-Yu Wang (4) , Liming Wu (1, 5)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 MODAL'X - Modélisation aléatoire de Paris X
4 SMS - School of Mathematical Sciences [Beijing]
5 Department of Mathematics Wuhan University

We continue our investigation on the transportation-information inequalities $W_pI$ for a symmetric markov process, introduced and studied in \cite{GLWY}. We prove that $W_pI$ implies the usual transportation inequalities $W_pH$, then the corresponding concentration inequalities for the invariant measure $\mu$. We give also a direct proof that the spectral gap in the space of Lipschitz functions for a diffusion process implies $W_1I$ (a result due to \cite{GLWY}) and a Cheeger type's isoperimetric inequality. Finally we exhibit relations between transportation-information inequalities and a family of functional inequalities (such as $\Phi$-log Sobolev or $\Phi$-Sobolev).

Mots clés: Wasserstein distance; entropy; Fisher information; transport-information inequality; deviation inequality

Domaines: Probabilités [math.PR]

Arnaud Guillin, Christian Léonard, Feng-Yu Wang, Liming Wu. Transportation-information inequalities for Markov processes (II) : relations with other functional inequalities. 2009. ⟨hal-00360854⟩