Results 1  10
of
22
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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Isoperimetry between exponential and Gaussian
 Electronic J. Prob
"... We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate halfspaces are approximate solutions of the isoperimetric problem. 1 ..."
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Cited by 16 (7 self)
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We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate halfspaces are approximate solutions of the isoperimetric problem. 1
INTERPOLATION BETWEEN LOGARITHMIC SOBOLEV AND POINCARÉ INEQUALITIES
"... Abstract. This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. 1. ..."
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Cited by 14 (6 self)
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Abstract. This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. 1.
ASYMPTOTICS OF THE FAST DIFFUSION EQUATION VIA ENTROPY ESTIMATES
"... Abstract. We consider nonnegative solutions of the fast diffusion equation ut = ∆u m with m ∈ (0, 1), in the Euclidean space R d, d ≥ 3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to t → ∞ for m ≥ mc = (d−2)/d, or as t approaches the extinction ti ..."
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Cited by 14 (7 self)
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Abstract. We consider nonnegative solutions of the fast diffusion equation ut = ∆u m with m ∈ (0, 1), in the Euclidean space R d, d ≥ 3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to t → ∞ for m ≥ mc = (d−2)/d, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a selfsimilar solution, for t large enough if m ≥ mc, or close enough to the extinction time if m < mc. Such results are new in the range m ≤ mc where previous approaches fail. In the range mc < m < 1 we improve on known results. 1.
Concentration for independent random variables with heavy tails
 AMRX
, 2005
"... If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of n independent copies, with good dependence in n. 1 ..."
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Cited by 14 (8 self)
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If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of n independent copies, with good dependence in n. 1
On the role of convexity in isoperimetry, spectralgap and concentration
 Invent. Math
"... We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the apriori weakest requirement that Lipschitz functions have arbitrarily slow uniform taildecay, are all quantitativ ..."
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Cited by 12 (3 self)
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We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the apriori weakest requirement that Lipschitz functions have arbitrarily slow uniform taildecay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov– Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “onaverage ” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst ” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan– Lovász–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semigroup following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvaturedimension condition of BakryÉmery. 1
An isoperimetric inequality for uniformly logconcave measures and uniformly convex bodies
, 2008
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A CHARACTERIZATION OF DIMENSION FREE CONCENTRATION IN TERMS OF TRANSPORTATION INEQUALITIES
, 2008
"... The aim of this paper is to show that a probability measure µ on R d concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand’s T2 transportationcost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villan ..."
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Cited by 7 (4 self)
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The aim of this paper is to show that a probability measure µ on R d concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand’s T2 transportationcost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villani. Generalizations to other types of concentration are also considered. In particular, one shows that the Poincaré inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of these results rely on simple Large Deviations techniques.
TRANSPORT INEQUALITIES. A SURVEY
"... Abstract. This is a survey of recent developments in the area of transport inequalities. We investigate their consequences in terms of concentration and deviation inequalities and sketch their links with other functional inequalities and also large deviation theory. ..."
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Cited by 7 (1 self)
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Abstract. This is a survey of recent developments in the area of transport inequalities. We investigate their consequences in terms of concentration and deviation inequalities and sketch their links with other functional inequalities and also large deviation theory.
Functional inequalities for heavy tails distributions and application to isoperimetry
, 2008
"... 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 th ..."
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Cited by 5 (4 self)
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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 onR 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.