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247
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.
Ricci curvature of Markov chains on metric spaces
 J. Funct. Anal
"... Abstract We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back, after scali ..."
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Cited by 85 (4 self)
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Abstract We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back, after scaling, the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the OrnsteinUhlenbeck process. Moreover this generalization is consistent with the BakryÉmery Ricci curvature for Brownian motion with a drift on a Riemannian manifold. Positive Ricci curvature is shown to imply a spectral gap, a LévyGromovlike Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. The bounds obtained are sharp in a variety of examples.
Comparison Geometry for the BakryEmery Ricci tensor
"... For Riemannian manifolds with a measure (M, g, e −f dvolg) we prove mean curvature and volume comparison results when the ∞BakryEmery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extension ..."
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Cited by 77 (7 self)
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For Riemannian manifolds with a measure (M, g, e −f dvolg) we prove mean curvature and volume comparison results when the ∞BakryEmery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the BakryEmery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results.
Convergence in distribution of random metric measure spaces (Λcoalescent measure trees)
, 2007
"... We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topol ..."
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Cited by 52 (10 self)
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We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows provided the sequence is tight from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultrametric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra)metric measure spaces given by the random genealogies of the Λcoalescents. We show that the Λcoalescent defines an infinite (random) metric measure space if and only if the socalled “dustfree”property holds.
Weak curvature conditions and functional inequalities
 J. of Funct. Anal
, 2007
"... Abstract. We give sufficient conditions for a measured length space (X, d, ν) to admit local and global Poincaré inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X, d, ν), defined in terms of transport of measures. We show that DM, together with a doubling conditi ..."
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Cited by 50 (2 self)
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Abstract. We give sufficient conditions for a measured length space (X, d, ν) to admit local and global Poincaré inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X, d, ν), defined in terms of transport of measures. We show that DM, together with a doubling condition on ν, implies a scaleinvariant local Poincaré inequality. We show that if (X, d, ν) has nonnegative NRicci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies DM, with constant 2 N. The condition DM is preserved by measured GromovHausdorff limits. We then prove a Sobolev inequality for measured length spaces with NRicci curvature bounded below by K> 0. Finally we derive a sharp global Poincaré inequality. There has been recent work on giving a good notion for a compact measured length space (X, d, ν) to have a “lower Ricci curvature bound”. In our previous work [10] we gave a notion of (X, d, ν) having nonnegative NRicci curvature, where N ∈ [1, ∞) is an effective dimension. The definition was in terms of the optimal transport of measures on X. A notion was also given of (X, d, ν) having ∞Ricci curvature bounded below by K ∈ R; a closely related definition in this case was given independently by Sturm [13]. In a recent
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
Curvaturedimension inequalities and Ricci lower bounds for subRiemannian manifolds with transverse symmetries
, 2012
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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 45 (12 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
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.
Heat flow on Alexandrov spaces
, 2012
"... We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the L²space produces the same evolution as the gradient flow of the relative entropy in the L²Wasserstein space. This means that the heat flow is well defined by either one of the t ..."
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Cited by 42 (15 self)
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We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the L²space produces the same evolution as the gradient flow of the relative entropy in the L²Wasserstein space. This means that the heat flow is well defined by either one of the two gradient flows. Combining properties of these flows, we are able to deduce the Lipschitz continuity of the heat kernel as well as BakryÉmery gradient estimates and the Γ2condition. Our identification is established by purely metric means, unlike preceding results relying on PDE techniques. Our approach generalizes to the case of heat flow with drift.