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43
ON THE EQUIVALENCE OF THE ENTROPIC CURVATUREDIMENSION CONDITION AND BOCHNER’S INEQUALITY ON METRIC MEASURE SPACES
, 2013
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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.
Optimal transport and Perelman’s reduced volume
, 2008
"... We show that a certain entropylike function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume. ..."
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Cited by 30 (2 self)
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We show that a certain entropylike function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume.
Dejan: Coarsening Rates for a Droplet Model: Rigorous Upper Bounds
"... Abstract. Certain liquids on solid substrates form a conguration of droplets connected by a precursor layer. This conguration coarsens: The average droplet size grows while the number of droplets decreases and the characteristic distance between them increases. We study this type of coarsening be ..."
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Cited by 25 (5 self)
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Abstract. Certain liquids on solid substrates form a conguration of droplets connected by a precursor layer. This conguration coarsens: The average droplet size grows while the number of droplets decreases and the characteristic distance between them increases. We study this type of coarsening behavior in a model given by an evolution equation for the lm height on an ndimensional substrate. Heuristic arguments based on the asymptotic analysis of Glasner and Witelski [6, 7] and numerical simulations suggest a statistically selfsimilar behavior characterized by a single exponent which determines the coarsening rate. In this paper, we establish rigorously an upper bound on the coarsening rate in a timeaveraged sense. We use the fact that the evolution is a gradient ow, i.e. a steepest descent in an energy landscape. Coarse information on the geometry of the energy landscape serves to obtain coarse information on the dynamics. This robust method was proposed in [10]. Our main analytical contribution is an interpolation inequality involving the Wasserstein distance which characterizes the coarse shape of the energy landscape. The upper bound we obtain is in agreement with heuristic arguments and numerical simulations. 1.
Ricci curvature of finite Markov chains via convexity of the entropy.
 Arch. Rational Mech. Anal.,
"... Abstract. We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the ..."
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Cited by 24 (6 self)
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Abstract. We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by BakryÉmery and OttoVillani. Furthermore we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.
Nonlinear mobility continuity equations and generalized displacement convexity
, 2009
"... We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the intern ..."
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Cited by 21 (4 self)
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We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As byproduct, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a nonrigorous argument indicating that they are not displacement semiconvex.
D.: Some Properties of Dirichlet LFunctions Associated with their Nontrivial Zeros I
"... Abstract. New results associated with the Extended Riemann Hypothesis on the zeros of the Dirichlet Lfunctions are obtained. The presentation of our work consists of two parts. In the present first part the connection between values of Lfunctions and Gauss sums is studied. This leads to a sufficie ..."
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Cited by 18 (4 self)
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Abstract. New results associated with the Extended Riemann Hypothesis on the zeros of the Dirichlet Lfunctions are obtained. The presentation of our work consists of two parts. In the present first part the connection between values of Lfunctions and Gauss sums is studied. This leads to a sufficient condition for the value s = 1 2 to be a zero of a given Lfunction. A necessary condition for the validity of the Extended Riemann Hypothesis is found. This involves the signs of the even derivatives of the analogue ξ(s, χ) of the wellknown ξ(s) function associated with the Riemann zetafunction. It is also proved that if the absolute value of a real even primitive character χ does not exceed the value 15 or the absolute value of a real odd primitive character χ does not exceed the value 7, then the value of the corresponding Lfunction and all its even derivatives at the point s = 1 2 are positive. In the second part of the presentation asymptotic formulas will be given for the even derivatives of the function ξ(s, χ) at s = 1 2 as the order of the derivative tends to infinity.
BakryÉmery curvaturedimension condition and Riemannian Ricci curvature bounds
, 2014
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On the BakryEmery criterion for linear diffusions and weighted porous media equations
 Comm. Math. Sci
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Interacting Diffusions Approximating the Porous Medium Equation and Propagation of Chaos
"... gemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. ..."
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Cited by 12 (0 self)
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gemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden.