Gradient flows of the entropy for finite Markov chains

Maas, Jan

doi:10.1016/j.jfa.2011.06.009

Cited by 272 publications

(443 citation statements)

References 27 publications

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“…One other famous strategy to establish transport inequalities in discrete setting is coming from the notion of curvature on discrete spaces introduced by Maas [Maa11]. Transport inequalities for invariant reversible measures of Markov chains are obtained from curvature type conditions (see also [EM12,EM14,EMT15]).…”

Section: Barycentric Transport Inequality and Logarithmic Sobolev Inementioning

confidence: 99%

“…Indeed, let us recall that the Talagrand's transport inequality T 2 is never satisfied by discrete measures. Recently, in the context of the study of curvature notion in discrete spaces, other transport inequalities have been proposed, mainly in the works by Erbar-Maas [Maa11,EM12]. However, due to the very abstract definition of the optimal transport costs, the associated concentration of measure phenomenon remains difficult to interpret.…”

Section: Introductionmentioning

confidence: 99%

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Concentration of Measure Principle and Entropy-Inequalities

Samson

2017

Convexity and Concentration

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Abstract. The concentration measure principle is presented in an abstract way to encompass and unify different concentration properties. We give a general overview of the links between concentration properties, transport-entropy inequalities, and logarithmic Sobolev inequalities for some specific transport costs. By giving few examples, we emphasize optimal weak transport costs as an efficient tool to establish new transport inequality and new concentration principles for discrete measures (the binomial law, the Poisson measure, the uniform law on the symmetric group).

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Section: Barycentric Transport Inequality and Logarithmic Sobolev Inementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Concentration of Measure Principle and Entropy-Inequalities

Samson

2017

Convexity and Concentration

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“…Another one initiated by the seminal work of Jordan et al [26] is to consider heat ow as the gradient ow of the relative entropy in the L -Wasserstein space. These interpretations and their equivalence were generalized to various settings ( [52], [39], [24], [18], [44], [35], [23], [10] etc.) including singular spaces without di erentiable structures.…”

Section: On the Curvature And Heat Flow On Hamiltonian Systemsmentioning

confidence: 99%

On the Curvature and Heat Flow on Hamiltonian Systems

Ohta

2014

Analysis and Geometry in Metric Spaces

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Abstract:We develop the di erential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation. We prove appropriate generalizations of the Bochner-Weitzenböck formula and Laplacian comparison theorem, and study the heat ow.Keywords: Hamiltonian, curvature, comparison theorem, heat ow MSC: Primary: 53C21; Secondary: 58J35, 49Q20 DOI 10.2478/agms-2014-0003 Received August 20, 2013 accepted February 14, 2014 IntroductionThe aim of this article is to apply the recently developed technique in Finsler geometry to the study of Hamiltonian systems. A Finsler manifold carries a (Minkowski) norm on each tangent space. Although Finsler manifolds form a much wider class than Riemannian manifolds, the notion of curvature makes sense and we can consider various comparison theorems similarly to the Riemannian case (see, e.g., [54]). Especially, the weighted Ricci curvature introduced by the author [41] has fruitful applications including the curvature-dimension condition ([41]), Laplacian comparison theorem for the natural nonlinear Laplacian ([44]), Bochner-Weitzenböck formula and gradient estimates ([46]), and generalizations of the CheegerGromoll splitting theorem ([43]). To be precise, the weighted Ricci curvature is de ned for a pair consisting of a Finsler manifold and a measure on it, and our Laplacian depends on the choice of the measure (see Subsections 2.3, 4.1 for details).Then, it is natural to expect that the theory of curvature can be applied beyond Finsler manifolds, and a class of manifolds M endowed with Lagragians L or, equivalently, Hamiltonians H is a natural choice. In fact, on the one hand, we know that Agrachev and Gamkrelidze [3] (see also [2]) have developed the theory of curvature operator for Hamiltonian systems in connection with optimal control theory (see [1] for a dynamical application). On the other hand, optimal transport theory (which is related to the curvature-dimension condition) for Lagrangian cost functions has been well investigated ([13], [19], [58]). Furthermore, Lee [31] recently showed a Riccati equation (see also [4], [5], [32] for the sub-Riemannian case) as well as convexity estimates for entropy functionals along smooth optimal transports for general (time-dependent) Hamiltonian systems, by means of the curvature operator. His uni ed approach recovers both the curvature-dimension condition (CD(K, ∞) and CD( , N) to be precise) for Riemannian or Finsler manifolds and various monotonicity formulas along ows in Riemannian metrics related to the Ricci ow.Our Hamiltonian will always be time-independent and non-negative. Compared with the Finsler situation, the lack of the homogeneity causes many di culties, for example, we need to take care of the di erence between the Lagrangian and the Hamiltonian (they coincide as functions via the Legendre transform in the Finsler case). Nevertheless, by combining the Riccati equation and the technique in the Finsler case, we prove ...

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“…The more representative class of examples is certainly given by the heat equation and its transformations. In different papers, covering different frameworks (see for instance [2,11,12,20,21,27,33,36,37,42,47]) it has been proved that curves of probability measures (µ t ) t≥0 with a density satisfying the adequate heat equation are exactly the curves with a speed equal to the opposite of the gradient of the relative Boltzmann entropy Ent, in the sense of optimal transport. Hence, we have the formal paradigm "μ t = −∇ Ent(µ t ) ⇐⇒ρ t = ∆ρ t " (1) where ρ t on the right-hand side denotes the density of µ t .…”

Section: Introductionmentioning

confidence: 99%

Diffusion by optimal transport in Heisenberg groups

Juillet

2013

Calc. Var.

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Abstract. We prove that the hypoelliptic diffusion of the Heisenberg group Hn describes, in the space of probability measures over Hn, a curve driven by the gradient flow of the Boltzmann entropy Ent, in the sense of optimal transport. We prove that conversely any gradient flow curve of Ent satisfy the hypoelliptic heat equation. This occurs in the subRiemannian Hn, which is not a space with a lower Ricci curvature bound in the metric sense of Lott-Villani and Sturm. Heisenberg group and optimal transport theory and gradient flow

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Gradient flows of the entropy for finite Markov chains

Cited by 272 publications

References 27 publications

Concentration of Measure Principle and Entropy-Inequalities

Concentration of Measure Principle and Entropy-Inequalities

On the Curvature and Heat Flow on Hamiltonian Systems

Diffusion by optimal transport in Heisenberg groups

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