On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities
On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities
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混合物关于测度浓度和索博列夫型不等式的精细性质
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
F. Malrieu
中科院分区：
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作者：
Djalil CHAFAÏ;F. Malrieu
混合物是法则的凸组合。尽管定义很简单，但混合物可能比其混合成分微妙得多。例如，混合高斯定律可能会产生多个深井的势能。我们在目前的工作中研究混合物关于测量浓度和索博列夫型函数不等式的精细性质。我们在一般混合物的情况下为 Lipschitz 函数提供尖锐的拉普拉斯界，涉及混合族的运输成本直径。此外，我们对二元混合物的 Sobolev 型不等式的分析揭示了与某种能带等周测量的自然关系，并支持通过质量传输进行约束插值。我们证明，当混合物比例达到 0 或 1 时，二元混合物的庞加莱常数可能保持有界，而对数索博列夫常数可能会令人惊讶地爆炸。这种反直觉的结果不能简化为支持断开连接，并且看起来像是两点空间上的方差-熵比较的回忆。就混合而言，对数 Sobolev 不等式不如 Poincar'e 不等式和 Lipschitz 函数的亚高斯集中稳定。我们在混凝土双组分混合物图库上展示了我们的结果。这项工作引出了许多悬而未决的问题。
Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed family. Additionally, our analysis of Sobolev type inequalities for two-component mixtures reveals natural relations with some kind of band isoperimetry and support constrained interpolation via mass transportation. We show that the Poincar'e constant of a two-component mixture may remain bounded as the mixture proportion goes to 0 or 1 while the logarithmic Sobolev constant may surprisingly blow up. This counter-intuitive result is not reducible to support disconnections, and appears as a reminiscence of the variance-entropy comparison on the two-point space. As far as mixtures are concerned, the logarithmic Sobolev inequality is less stable than the Poincar'e inequality and the sub-Gaussian concentration for Lipschitz functions. We illustrate our results on a gallery of concrete two-component mixtures. This work leads to many open questions.