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

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

2009-02-12

arxiv.org/abs/0902.2101v1

Mathematics

Probability

Scientific paper

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).

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