Functional inequalities for heavy tailed distributions and application to isoperimetry
Patrick Cattiaux, Institut Mathématique de Toulouse
Nathael Gozlan, Université Marne La Vallée
Arnaud Guillin, Université Blaise Pascal
Cyril Roberto, Université Marne La Vallée
Abstract
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 result
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