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Analytic and Geometric Logarithmic Sobolev Inequalities

Michel Ledoux — 2011

Journées Équations aux dérivées partielles

We survey analytic and geometric proofs of classical logarithmic Sobolev inequalities for Gaussian and more general strictly log-concave probability measures. Developments of the last decade link the two approaches through heat kernel and Hamilton-Jacobi equations, inequalities in convex geometry and mass transportation.

Sobolev-Kantorovich Inequalities

Michel Ledoux — 2015

Analysis and Geometry in Metric Spaces

In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means...

On Talagrand's deviation inequalities for product measures

Michel Ledoux — 2010

ESAIM: Probability and Statistics

We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M. Talagrand in the recent years. Our method is based on functional inequalities of Poincaré and logarithmic Sobolev type and iteration of these inequalities. In particular, we establish with theses tools sharp deviation inequalities from the mean on norms of sums of independent random vectors and empirical processes. Concentration for the Hamming distance may also be deduced...

A logarithmic Sobolev form of the Li-Yau parabolic inequality.

Dominique BakryMichel Ledoux — 2006

Revista Matemática Iberoamericana

We present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of non-negatively curved diffusion operators that contains and improves upon the Li-Yau parabolic inequality. This new inequality is of interest already in Euclidean space for the standard Gaussian measure. The result may also be seen as an extended version of the semigroup commutation properties under curvature conditions. It may be applied to reach optimal Euclidean logarithmic Sobolev inequalities...

On Measure Concentration of Vector-Valued Maps

Michel LedouxKrzysztof Oleszkiewicz — 2007

Bulletin of the Polish Academy of Sciences. Mathematics

We study concentration properties for vector-valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in k . To this end, we study in particular a domination principle for projections which might be of independent interest. We further compare our conclusions with earlier results by Pinelis in the Gaussian case, and discuss extensions to the infinite-dimensional setting.

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