Transportation-information inequalities for Markov processes (II) : relations with other functional inequalities
Transportation-information inequalities for Markov processes (II) : relations with other functional inequalities
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发表时间：
2009-02
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通讯作者：
A. Guillin;C. Léonard;Feng-Yu Wang;Liming Wu
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作者：
A. Guillin;C. Léonard;Feng-Yu Wang;Liming Wu
我们继续我们的调查运输信息不等式$W_pI$的对称马尔可夫过程，介绍和研究\cite{GLWY}。我们证明了$W_pI$蕴含通常的运输不等式$W_pH$，然后对应不变测度$\mu$的浓度不等式。本文还直接证明了扩散过程的Lipschitz函数空间的谱间隙蕴含W_1I（GLWY的一个结果）和Cheeger型等周不等式。最后，我们展示了运输信息不等式和一个家庭的功能不等式（如$\Phi$-log Sobolev或$\Phi$-Sobolev）之间的关系。
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).