Wasserstein geometry of porous medium equation

Takatsu, Asuka

doi:10.1016/j.anihpc.2011.10.003

Cited by 9 publications

(13 citation statements)

References 21 publications

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“…In [26,29], the authors show that a similar statement holds for the porous medium equation on the space of q-Gaussian measures. That is, a solution to the porous medium equation with an initial data being a q-Gaussian measure is again a q-Gaussian for all time.…”

Section: Q-gaussian Measures and Solutions Of The Porous Medium Equationmentioning

confidence: 72%

“…The q-exponential function and its inverse, the q-logarithmic function, are defined respectively by [29] exp…”

Section: Resultsmentioning

confidence: 99%

“…We recall relevant properties of the q-Gaussian measures as given in [29]. The q-exponential function and q-logarithmic function satisfy the following properties…”

Section: Properties Of Q-gaussian Measuresmentioning

confidence: 99%

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Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation

Duong

2015

Asymptotic Analysis

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In this paper, we study the Wasserstein gradient flow structure of the porous medium equation restricted to q-Gaussians. The JKO-formulation of the porous medium equation gives a variational functional K h , which is the sum of the (scaled-) Wasserstein distance and the internal energy, for a time step h. We prove that, for the case of q-Gaussians on the real line, K h is asymptotically equivalent, in the sense of Γ -convergence as h tends to zero, to a rate-large-deviation-like functional. The result explains why the Wasserstein metric as well as the combination of it with the internal energy play an important role.1 E 1 is the Boltzmann-Shannon entropy. 0921-7134/15/$35.00 © 2015 -IOS Press and the authors. All rights reserved 86 M.H. Duong / Asymptotic equivalence of the discrete variational functional with respect to the Wasserstein distance W 2 on the space of probability measures with finite second moment P 2 (R d ),where Γ (μ, ν) is the set of all probability measures on R d × R d with marginals μ and ν. This statement can be understood in a variety of different ways. In [27], Otto shows this from a differential geometry point of view by considering (P 2 (R d ), W 2 ) as an infinite dimensional Riemannian manifold. However, for the purpose of this paper, the most useful way is that the solution t → ρ(t, x) can be approximated by the JKO-discretization scheme as follows [17,31].

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Section: Q-gaussian Measures and Solutions Of The Porous Medium Equationmentioning

confidence: 72%

“…The q-exponential function and its inverse, the q-logarithmic function, are defined respectively by [29] exp…”

Section: Resultsmentioning

confidence: 99%

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Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation

Duong

2015

Asymptotic Analysis

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“…We stress that N becomes negative for m > 1, then Ric N is defined in the same form as the case of N ∈ (n, ∞) (see Definition 2.1). Similarly to CD(K, ∞), we can derive from Hess H m ≥ K > 0 the associated functional inequalities (see also [AGK], [CGH], [Ta1] for related works) and the concentration of measures (in terms of exp m ). Furthermore, the gradient flow of H m produces weak solutions to the fast diffusion equation (m < 1) or the porous medium equation (m > 1) with drift of the form…”

Section: Introductionmentioning

confidence: 99%

Displacement convexity of generalized relative entropies. II

Ohta

Takatsu

2013

Communications in Analysis and Geometry

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We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the nonnegative weighted Ricci curvature and the convexity of another weight function used in the definition of the generalized relative entropies. This convexity condition corresponds to Lott and Villani's version of the curvature-dimension condition. As applications, we obtain appropriate variants of the Talagrand, HWI and logarithmic Sobolev inequalities, as well as the concentration of measures. We also investigate the gradient flow of our generalized relative entropy.

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“…Gradient flow from the view of Wasserstein geometry has been investigated by Otto [Ot] in the Euclidean case, and by Villani [Vi2,Chapters 23,24] in the weighted Riemannian case in a different manner from ours. Functional inequalities related to the convexity of the weight Ψ were studied in [AGK], [CGH] and [Ta2] in Euclidean spaces (see also [St1,Remark 1.1] and [Vi2,Chapters 24,25]). The concentration of measures seems new even in the Euclidean setting.…”

Section: Introductionmentioning

confidence: 99%

Displacement convexity of generalized relative entropies

Ohta

Takatsu

2011

Advances in Mathematics

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We investigate the m-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement K-convexity of the m-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the K-convexity of the weight function. We use this to show appropriate variants of the Talagrand, HWI and the logarithmic Sobolev inequalities, as well as the concentration of measures. We also prove that the gradient flow of the m-relative entropy produces a solution to the porous medium equation or the fast diffusion equation.

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Wasserstein geometry of porous medium equation

Cited by 9 publications

References 21 publications

Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation

Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation

Displacement convexity of generalized relative entropies. II

Displacement convexity of generalized relative entropies

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