[Structure de Finsler dans lʼespace de Wasserstein et flux de gradient]
Il est connu que lʼespace des mesures de probabilités muni de la distance de Wasserstein (lʼespace de Wasserstein ) est une variété Riemanienne (voir F. Otto (2001) [9]). Ici, nous montrons que lorsquʼon change le coût quadratique en un coût plus general, homogène de degré , lʼespace correspondant (lʼespace de Wasserstein ) admet une structure de Finsler dont la distance induite est la distance de Wasserstein . Grâce à cette structure de Finsler, nous donnons une définition de la différentiel et du gradient des fonctionelles définies sur cet espace, et aussi des flux de gradient sur cet espace. En particulier nous montrons que lʼéquation parabolique q-Laplacien est un flux de gradient dans lʼespace de Wasserstein pour . Quand , nous retrouvons la structure Remannienne de F. Otto, ce qui confirme que lʼespace de Wasserstein est une variété Riemanienne de Finsler. Notre méthode sʼapplique à des mesures de probabilité absolument continues par rapport à la mesure de Lebesgue dans , et dont les densités sont strictement positives.
It is known from the work of F. Otto (2001) [9], 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 , 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 . When , 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 , and they have smooth and strictly positive densities.
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@article{CRMATH_2012__350_1-2_35_0,
author = {Agueh, Martial},
title = {Finsler structure in the {\protect\emph{p}-Wasserstein} space and gradient flows},
journal = {Comptes Rendus. Math\'ematique},
pages = {35--40},
publisher = {Elsevier},
volume = {350},
number = {1-2},
year = {2012},
doi = {10.1016/j.crma.2011.11.014},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.11.014/}
}
TY - JOUR AU - Agueh, Martial TI - Finsler structure in the p-Wasserstein space and gradient flows JO - Comptes Rendus. Mathématique PY - 2012 SP - 35 EP - 40 VL - 350 IS - 1-2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.11.014/ DO - 10.1016/j.crma.2011.11.014 LA - en ID - CRMATH_2012__350_1-2_35_0 ER -
%0 Journal Article %A Agueh, Martial %T Finsler structure in the p-Wasserstein space and gradient flows %J Comptes Rendus. Mathématique %D 2012 %P 35-40 %V 350 %N 1-2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.11.014/ %R 10.1016/j.crma.2011.11.014 %G en %F CRMATH_2012__350_1-2_35_0
Agueh, Martial. Finsler structure in the p-Wasserstein space and gradient flows. Comptes Rendus. Mathématique, Tome 350 (2012) no. 1-2, pp. 35-40. doi : 10.1016/j.crma.2011.11.014. http://www.numdam.org/articles/10.1016/j.crma.2011.11.014/
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