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231
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.
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.
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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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.
ON THE EQUIVALENCE OF THE ENTROPIC CURVATUREDIMENSION CONDITION AND BOCHNER’S INEQUALITY ON METRIC MEASURE SPACES
, 2013
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RICCI FLOW, ENTROPY AND OPTIMAL TRANSPORTATION
"... Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented ndimensional manifold M. Suppose two families of normalized nforms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ..."
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Cited by 37 (1 self)
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Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented ndimensional manifold M. Suppose two families of normalized nforms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ∆g(τ). If these nforms represent two evolving distributions of particles over M, the minimum rootmeansquare distance W2(ω(τ), ˜ω(τ), τ) to transport the particles of ω(τ) onto those of ˜ω(τ) is shown to be nonincreasing as a function of τ, without sign conditions on the curvature of (M, g(τ)). Moreover, this contractivity property is shown to characterize supersolutions to the Ricci flow.
Eulerian calculus for the displacement convexity in the Wasserstein distance
, 2008
"... In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals de ned on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifo ..."
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Cited by 34 (4 self)
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In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals de ned on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by OttoWestdickenberg in [19] and on the metric characterization of the gradient ows generated by the functionals in the Wasserstein space.
On the Heat flow on metric measure spaces: existence, uniqueness and stability
, 2009
"... We prove existence and uniqueness of the gradient flow of the Entropy functional under the only assumption that the functional is λgeodesically convex for some λ ∈ R. Also, we prove a general stability result for gradient flows of geodesically convex functionals which Γ−converge to some limit funct ..."
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Cited by 31 (12 self)
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We prove existence and uniqueness of the gradient flow of the Entropy functional under the only assumption that the functional is λgeodesically convex for some λ ∈ R. Also, we prove a general stability result for gradient flows of geodesically convex functionals which Γ−converge to some limit functional. The stability result applies directly to the case of the Entropy functionals on compact spaces. 1