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Finsler structure in the \(p\)-Wasserstein space and gradient flows. (English. Abridged French version) Zbl 1239.53021

Author’s abstract: It is known from the work of F. Otto [Commun. Partial Differ. Equations 26, No. 1–2, 101–174 (2001; Zbl 0984.35089)], that the space of probability measures equipped with the quadratic Wasserstein distance, i.e., the 2-Wasserstein space, can be viewed as a Riemannian manifold. Here we show that when the quadratic cost is replaced by a general homogeneous cost of degree \(p>1\), the corresponding space of probability measures, i.e., the \(p\)-Wasserstein space, can be endowed with a Finsler metric whose induced distance function is the \(p\)-Wasserstein distance. Using this Finsler structure of the \(p\)-Wasserstein space, we give definitions of the differential and gradient of functionals defined on this space, and then of gradient flows in this space. In particular we show in this framework that the parabolic \(q\)-Laplacian equation is a gradient flow in the \(p\)-Wasserstein space, where \(p=q/(q - 1)\). When \(p=2\), we recover the Riemannian structure introduced by F. Otto, which confirms that the 2-Wasserstein space is a Riemann-Finsler manifold. Our approach is confined to a smooth situation where probability measures are absolutely continuous with respect to the Lebesgue measure on \(\mathbb R^n\), and they have smooth and strictly positive densities.
Reviewer’s remark: Having this structure, the total space of the tangent bundle can be endowed with a non-linear connection...

MSC:

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
28A33 Spaces of measures, convergence of measures
76S05 Flows in porous media; filtration; seepage
35Q35 PDEs in connection with fluid mechanics

Citations:

Zbl 0984.35089
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References:

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