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445
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.
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 115 (32 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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Continuity, curvature, and the general covariance of optimal transportation
, 2008
"... Let M and ¯ M be ndimensional manifolds equipped with suitable Borel probability measures ρ and ¯ρ. For subdomains M and ¯ M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4 (M × ¯ M) to guarantee smoothness of the optimal map pushing ρ forward to ¯ρ; the ..."
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Cited by 75 (20 self)
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Let M and ¯ M be ndimensional manifolds equipped with suitable Borel probability measures ρ and ¯ρ. For subdomains M and ¯ M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4 (M × ¯ M) to guarantee smoothness of the optimal map pushing ρ forward to ¯ρ; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via nonnegativity of the sectional curvature of certain nullplanes in a novel but natural pseudoRiemannian geometry which the cost c induces on the product space M × ¯ M. We also explore some connections between optimal transportation and spacelike Lagrangian submanifolds in symplectic geometry. Using the pseudoRiemannian structure, we extend Ma, Trudinger and Wang’s conditions to transportation costs on differentiable manifolds, and provide a direct elementary proof of a maximum principal characterizing it due to Loeper, relaxing his hypotheses even for subdomains M and ¯ M of Rn. This maximum principle plays a key role in Loeper’s Hölder continuity theory of optimal maps. Our proof allows his theory to be made logically independent of all earlier works, and sets the stage for extending it to new global settings, such as general submersions and tensor products of the specific Riemannian manifolds he considered.
Globalintime weak measure solutions, finitetime aggregation and confinement for nolocal interaction equations
, 2009
"... In this paper, we provide a wellposedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of ..."
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Cited by 68 (19 self)
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In this paper, we provide a wellposedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. We develop an existence theory that enables one to go beyond the blowup time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, we show the finite time total collapse of the solution onto a single point, for compactly supported initial measures. Finally, we give conditions on compensation between the attraction at large distances and local repulsion of the potentials to have globalintime confined systems for compactly supported initial data. Our approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations.
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 54 (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.
Heat flow on Finsler manifolds
, 2009
"... This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : TxM → R+ on each tangent space. Mostly, we will require that this norm is strongly convex and smooth and that it depends smoothly on the base point x. The particu ..."
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Cited by 47 (18 self)
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This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : TxM → R+ on each tangent space. Mostly, we will require that this norm is strongly convex and smooth and that it depends smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold: • either as gradient flow on L2 (M,m) for the energy E(u) = 1
EULERIAN CALCULUS FOR THE CONTRACTION IN THE WASSERSTEIN DISTANCE
"... We consider the porous medium equation on a compact Riemannian manifold and give a new proof of the contraction of its semigroup in the Wasserstein distance. This proof is based on the insight that the porous medium equation does not increase the size of infinitesimal perturbations along gradient fl ..."
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Cited by 43 (4 self)
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We consider the porous medium equation on a compact Riemannian manifold and give a new proof of the contraction of its semigroup in the Wasserstein distance. This proof is based on the insight that the porous medium equation does not increase the size of infinitesimal perturbations along gradient flow trajectories, and on an Eulerian formulation for the Wasserstein distance using smooth curves. Our approach avoids the existence result for optimal transport maps on Riemannian manifolds.