Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry

By Patrick Cattiaux, Nathael Gozlan, Arnaud Guillin, Cyril Roberto. 2010

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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 Poincare and weak Cheeger, weighted Poincare and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures on $\mathbb{R}^n$ 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

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7641a392bd8afec126e13c7928607307f48a1565

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Poincaré inequalities and hitting times (2010, methodology): “…function ϕ growing to inﬁnity and R > 0, such that ϕ v q (x) ≤ qv q − 1 (x) for | x | ≥ R. (5.9) Indeed if (5.9) holds, we have Lv q (x) ≤ − ϕ v q (x) for | x | large enough, and v q is thus a ϕ -Lyapunov function in the terminology of [12] and [19] (see Deﬁnition 2.2 in the ﬁrst reference).”
Central limit theorems for additive functionals of ergodic Markov diffusions processes (2011, unknown): “ αdx associated to the να-reversible generator L= ∆−αx|x|α−2.∇. With W(x) = ea|x|α for large |x|, one easily gets that for large |x| LW(x) ≤ −cα2a(a−1)|x|2α−2ea|x|α so that by Theorems 2.8 and 5.1 in [CGGR10], we get that να veriﬁes a weak Poincar´e inequality with β(s) = kn,α log(2/s) 2 α −2. We ma”
A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality (2008, background): “One will also remark that contrary to [5, 12 ,13], we will not use localization technique, constants are thus easier to derive.”