Results 11  20
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131
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.
A family of nonlinear fourth order equations of gradient flow type
, 2009
"... Global existence and longtime behavior of solutions to a family of nonlinear fourth order evolution equations on Rd are studied. These equations constitute gradient flows for the perturbed information functionals Fα,λ(u) = 1 ..."
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Cited by 33 (9 self)
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Global existence and longtime behavior of solutions to a family of nonlinear fourth order evolution equations on Rd are studied. These equations constitute gradient flows for the perturbed information functionals Fα,λ(u) = 1
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.
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.
Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
, 2005
"... We investigate PrékopaLeindler type inequalities on a Riemannian manifold M equipped with a measure with density e−V where the potential V and the Ricci curvature satisfy Hessx V + Ricx ≥ λ I for all x ∈ M, with some λ ∈ R. As in our earlier work [14], the argument uses optimal mass transport on M, ..."
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Cited by 23 (2 self)
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We investigate PrékopaLeindler type inequalities on a Riemannian manifold M equipped with a measure with density e−V where the potential V and the Ricci curvature satisfy Hessx V + Ricx ≥ λ I for all x ∈ M, with some λ ∈ R. As in our earlier work [14], the argument uses optimal mass transport on M, but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method to logarithmic Sobolev inequalities (the BakryEmery criterion will be recovered) and to transport inequalities. A study of the displacement convexity of the entropy functional completes the exposition.
Convergence to equilibrium in Wasserstein distance for FokkerPlanck equations
 J. Funct. Anal
"... We describe conditions on nongradient drift diffusion FokkerPlanck equations for its solutions to converge to equilibrium with a uniform exponential rate in Wasserstein distance. This asymptotic behaviour is related to a functional inequality, which links the distance with its dissipation and en ..."
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Cited by 22 (8 self)
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We describe conditions on nongradient drift diffusion FokkerPlanck equations for its solutions to converge to equilibrium with a uniform exponential rate in Wasserstein distance. This asymptotic behaviour is related to a functional inequality, which links the distance with its dissipation and ensures a spectral gap in Wasserstein distance. We give practical criteria for this inequality and compare it to classical ones. The key point is to quantify the contribution of the diffusion term to the rate of convergence, in any dimension, which to our knowledge is a novelty.
Poissontype deviation inequalities for curved continuous time Markov chains
 Bernoulli 13 (2007), n
"... In this paper, we present new Poissontype deviation inequalities for continuous time Markov chains whose Wasserstein curvature or Γcurvature is bounded below. Although these two curvatures are equivalent for Brownian motion on Riemannian manifolds, they are not comparable in discrete settings and ..."
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Cited by 20 (3 self)
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In this paper, we present new Poissontype deviation inequalities for continuous time Markov chains whose Wasserstein curvature or Γcurvature is bounded below. Although these two curvatures are equivalent for Brownian motion on Riemannian manifolds, they are not comparable in discrete settings and yield different deviation bounds. In the case of birthdeath processes, we provide some conditions on the transition rates of the associated generator for such curvatures to be bounded below, and we extend the deviation inequalities established by Ané and Ledoux (2000) for continuous time random walks, seen as models in null curvature. Some applications of these tail estimates are given to Brownian driven OrnsteinUhlenbeck processes and M/M/1 queues.