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446
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 230 (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.
Contractions in the 2Wasserstein Length Space and Thermalization of Granular Media
, 2004
"... An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical ..."
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Cited by 114 (35 self)
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An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinitedimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even nonconvexity) — if uniformly controlled — will quantify contractivity (limit expansivity) of the flow.
Blowup in multidimensional aggregation equations with mildly singular interaction kernels
 Nonlinearity
, 2009
"... interaction kernels ..."
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On the second boundary value problem for MongeAmpère type equations and optimal
"... Abstract. This paper is concerned with the existence of globally smooth solutions for the second boundary value problem for MongeAmpère type equations and the application to regularity of potentials in optimal transportation. The cost functions satisfy a weak form of the condition A3, which was int ..."
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Cited by 73 (4 self)
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Abstract. This paper is concerned with the existence of globally smooth solutions for the second boundary value problem for MongeAmpère type equations and the application to regularity of potentials in optimal transportation. The cost functions satisfy a weak form of the condition A3, which was introduced in a recent paper with Xinan Ma in conjunction with interior regularity. Consequently they include the quadratic cost function case of Caffarelli and Urbas as well as the various examples in the earlier work. The approach is through the derivation of global estimates for second derivatives of solutions. 1.
Asymptotic Flocking Dynamics for the kinetic CuckerSmale model
, 2009
"... Abstract. In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting ..."
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Cited by 67 (14 self)
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Abstract. In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting from a Boltzmanntype equation. The largetime behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [16] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution.
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 58 (8 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.
Convergence of the masstransport steepest descent scheme for the subcritical PatlakKellerSegel model
 SIAM J. Numer. Anal
"... Abstract. Variational steepest descent approximation schemes for the modified PatlakKellerSegel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean W ..."
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Cited by 56 (19 self)
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Abstract. Variational steepest descent approximation schemes for the modified PatlakKellerSegel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated to this equation for subcritical masses. As a consequence, we recover the recent result about the global in time existence of weaksolutions to the modified PatlakKellerSegel equation for the logarithmic interaction kernel in any dimension in the subcritical case. Moreover, we show how this method performs numerically in one dimension. In this particular case, this numerical scheme corresponds to a standard implicit Euler method for the pseudoinverse of the cumulative distribution function. We demonstrate its capabilities to reproduce easily without the need of meshrefinement the blowup of solutions for supercritical masses. 1.
Functional inequalities, thick tails and asymptotics for the critical mass PatlakKellerSegel model
, 2011
"... We investigate the long time behavior of the critical mass PatlakKellerSegel equation. This equation has a one parameter family of steadystate solutions λ, λ> 0, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attrac ..."
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Cited by 51 (12 self)
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We investigate the long time behavior of the critical mass PatlakKellerSegel equation. This equation has a one parameter family of steadystate solutions λ, λ> 0, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attraction for them using an entropy functional Hλ coming from the critical fast diffusion equation in R 2. We construct solutions of PatlakKellerSegel equation satisfying an entropyentropy dissipation inequality for Hλ. While the entropy dissipation for Hλ is strictly positive, it turns out to be a difference of two terms, neither of which need to be small when the dissipation is small. We introduce a strategy of controlled concentration to deal with this issue, and then use the regularity obtained from the entropyentropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards λ. In the present paper, we do not provide any estimate of the rate of convergence, but we discuss how this would result from a stability result for a certain sharp GagliardoNirenbergSobolev inequality.
Optimal mass transportation and Mather theory
 JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
, 2005
"... We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz ..."
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Cited by 51 (4 self)
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We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is absolutely continuous. We explain the relations with Mather’s minimal measures.