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

TL;DR

This paper explores the relationships between transportation-information inequalities and other functional inequalities for Markov processes, establishing implications for concentration, spectral gaps, and isoperimetric inequalities.

Contribution

It demonstrates that $W_pI$ implies $W_pH$ and concentration inequalities, and links transportation-information inequalities with spectral gaps and isoperimetric inequalities.

Findings

$W_pI$ implies $W_pH$ and concentration inequalities

Spectral gap implies $W_1I$ for diffusion processes

Relations established between transportation-information and functional inequalities

Abstract

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).