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

Abstract

We continue our investigation on the transportation-information inequalities $W_{p}I$ for a symmetric markov process, introduced and studied in \cite{GLWY}. We prove that $W_{p}I$ implies the usual transportation inequalities $W_{p}H$, 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 $W_{1}I$ (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).

References in corpus (1)

• Transportation cost-information inequalities and applications to random dynamical systems and diffusions