Functional inequalities for heavy tails distributions and application to isoperimetry

Patrick Cattiaux (1) , Nathael Gozlan (2) , Arnaud Guillin (3) , Cyril Roberto (2)

1 LSP - Laboratoire de Statistique et Probabilités
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 LATP - Laboratoire d'Analyse, Topologie, Probabilités

This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures on $\R^n$ we obtain the optimal dimension dependence using the mass transportation method. Then we derive (optimal) isoperimetric inequalities. Finally we deal with spherically symmetric measures. We recover and improve many previous results.

Mots clés: weighted Poincaré inequalities; weighted Cheeger inequalities; Lyapunov function; weak inequalities; isoperimetric profile

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Patrick Cattiaux, Nathael Gozlan, Arnaud Guillin, Cyril Roberto. Functional inequalities for heavy tails distributions and application to isoperimetry. Electronic Journal of Probability, 2010, 15 (13), pp.364-385. ⟨10.1214/EJP.v15-754⟩. ⟨hal-00300766⟩