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124
Ricci curvature for metricmeasure spaces via optimal transport
 ANN. OF MATH
, 2005
"... We define a notion of a measured length space X having nonnegative NRicci curvature, for N ∈ [1, ∞), or having ∞Ricci curvature bounded below by K, for K ∈ R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of proba ..."
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Cited by 231 (10 self)
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We define a notion of a measured length space X having nonnegative NRicci curvature, for N ∈ [1, ∞), or having ∞Ricci curvature bounded below by K, for K ∈ R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured GromovHausdorff limits. We give geometric and analytic consequences.
A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
, 2001
"... A concavity estimate is derived for interpolations between L¹(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian ..."
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Cited by 83 (9 self)
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A concavity estimate is derived for interpolations between L¹(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian versions of these theorems incorporate a volume distortion factor which can, however, be controlled via lower bounds on Ricci curvature. The method uses optimal mappings from mass transportation theory. Along the way, several new properties are established for optimal mass transport and interpolating maps on a Riemannian manifold.
Transportation costinformation inequalities and applications to random dynamical systems and diffusions
 ANN. PROBAB
, 2004
"... We first give a characterization of the L 1transportation costinformation inequality on a metric space and next find some appropriate sufficient condition to transportation costinformation inequalities for dependent sequences. Applications to random dynamical systems and diffusions are studied. ..."
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Cited by 60 (9 self)
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We first give a characterization of the L 1transportation costinformation inequality on a metric space and next find some appropriate sufficient condition to transportation costinformation inequalities for dependent sequences. Applications to random dynamical systems and diffusions are studied.
Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below
, 2006
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HARNACK INEQUALITY AND APPLICATIONS FOR STOCHASTIC GENERALIZED POROUS MEDIA EQUATIONS
 SUBMITTED TO THE ANNALS OF PROBABILITY
, 2006
"... By using coupling and Girsanov transformations, the dimensionfree Harnack inequality and the strong Feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations. As applications, explicit upper bounds of the Lpnorm of the density as ..."
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Cited by 37 (10 self)
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By using coupling and Girsanov transformations, the dimensionfree Harnack inequality and the strong Feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations. As applications, explicit upper bounds of the Lpnorm of the density as well as hypercontractivity, ultracontractivity and compactness of the corresponding semigroup are derived.
inequalities on manifolds with boundary and applications
 J. Math. Pures Appl
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Harnack Inequality and Strong Feller Property for Stochastic FastDiffusion Equations
, 2007
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Transportationinformation inequalities for Markov processes (II): Relations with other functional inequalities
, 2009
"... We continue our investigation on the transportationinformation inequalities WpI for a symmetric markov process, introduced and studied in [13]. We prove that WpI implies the usual transportation inequalities WpH, then the corresponding concentration inequalities for the invariant measure µ. We giv ..."
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Cited by 31 (10 self)
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We continue our investigation on the transportationinformation inequalities WpI for a symmetric markov process, introduced and studied in [13]. We prove that WpI implies the usual transportation inequalities WpH, then the corresponding concentration inequalities for the invariant measure µ. We give also a direct proof that the spectral gap in the space of Lipschitz functions for a diffusion process implies W1I (a result due to [13]) and a Cheeger type’s isoperimetric inequality. Finally we exhibit relations between transportationinformation inequalities and a family of functional inequalities (such as Φlog Sobolev or ΦSobolev).
Convergence to equilibrium for granular media equations and their Euler schemes
 Ann. Appl. Probab
"... We introduce a new interacting particle system to investigate the behavior of the nonlinear, nonlocal diffusive equation already studied by Benachour et al in [3, 4]. We first prove an uniform (with respect to time) propagation of chaos. Then, we show that the solution of the nonlinear PDE converge ..."
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Cited by 29 (4 self)
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We introduce a new interacting particle system to investigate the behavior of the nonlinear, nonlocal diffusive equation already studied by Benachour et al in [3, 4]. We first prove an uniform (with respect to time) propagation of chaos. Then, we show that the solution of the nonlinear PDE converges exponentially fast to equilibrium recovering a result established by an other way by Carrillo et al [7]. At last we provide explicit and Gaussian confidence intervals for the convergence of an implicit Euler scheme to the stationary distribution of the nonlinear equation. 1. Introduction In this paper, we study an interacting particle system and an implicit Euler scheme to describe and solve numerically, by a probabilistic way,