On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities

Chafaï, Djalil; Malrieu, Florent

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1, pp. 72-96.

Les mélanges dont il est question ici sont des combinaisons convexes de lois de probabilité. Malgré cette définition simple, un mélange peut être beaucoup plus subtil que ses composants. Un mélange de lois gaussiennes par exemple peut donner lieu à des potentiels à profonds puits multiples. Dans ce travail, nous étudions les propriétés fines des mélanges vis à vis de la concentration de la mesure et des inégalités de type Sobolev. Nous proposons des bornes sur la transformée de Laplace faisant intervenir le diamètre de la famille mélangée pour une distance de transport. Notre analyse des inégalités de type Sobolev pour les mélanges à deux composants révèle des relations naturelles avec une forme d'isopérimétrie pour les bandes, ainsi qu'avec le transport optimal sous contrainte de support. Nous établissons que la constante de Poincaré peut rester bornée lorsque la proportion du mélange tend vers 0 tandis que la constante de Sobolev logarithmique peut exploser. Ce phénomène contre intuitif n'est pas réductible à un problème de support et peut être vu comme une trace de la comparaison variance-entropie sur l'espace à deux points. Pour les mélanges, la propriété de concentration de la mesure sous-gaussienne et l'inégalité de Poincaré sont plus stables que l'inégalité de Sobolev logarithmique. Nous illustrons nos résultats avec une collection d'exemples à deux composants concrets. Ce travail conduit à plusieurs questions ouvertes.

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 Poincaré 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 Poincaré 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.

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