On Monge–Kantorovich problem in the plane

Shen, Yinfang; Zheng, Weian

doi:10.1016/j.crma.2009.11.022

Partial Differential Equations/Probability Theory

On Monge–Kantorovich problem in the plane
[Sur le problème de Monge–Kantorovich du plan]

Présenté par : Marc Yor

Shen, Yinfang ^1,² ; Zheng, Weian ^1,²

¹ SFS, ITCS, East China Normal University, Shanghai, China, 200062
² Department of Mathematics, University of California, Irvine, CA 92697, USA

Comptes Rendus. Mathématique, Tome 348 (2010) no. 5-6, pp. 267-271.

Résumé
Abstract

Nous utilisons une méthode probabiliste pour transformer le célèbre problème de Monge–Kantorovich dans une région bornée du plan Euclidien à celui de Dirichlet associé à une équation aux dérivées partielles quasi-linéaire :

\frac{\partial}{\partial x} A (x, F_{x}^{'}) + \frac{\partial}{\partial y} B (y, F_{y}^{'}) = 0

où

A_{y}^{'} (., .) > 0, B_{x}^{'} (., .) > 0

, et F est une loi de probabilité inconnue. Ainsi, nous avons développé une nouvelle méthode probabiliste pour l'équation de Monge–Ampère associé au problème ci-dessus.

We use a simple probability method to transform the celebrated Monge–Kantorovich problem in a bounded region of Euclidean plane into a Dirichlet boundary problem associated to a quasi-linear elliptic equation with 0-order term missing in its diffusion coefficients:

\frac{\partial}{\partial x} A (x, F_{x}^{'}) + \frac{\partial}{\partial y} B (y, F_{y}^{'}) = 0

where

A_{y}^{'} (., .) > 0, B_{x}^{'} (., .) > 0

and F is an unknown probability distribution function. Thus, we are able to give a probability approach to the famous Monge–Ampère equation, which is known to be associated to the above problem.

Reçu le : 2008-08-19
Accepté le : 2009-11-02
Publié le : 2010-02-20

DOI : 10.1016/j.crma.2009.11.022

Affiliations des auteurs :

Shen, Yinfang ^{1, 2} ; Zheng, Weian ^{1, 2}

¹ SFS, ITCS, East China Normal University, Shanghai, China, 200062
² Department of Mathematics, University of California, Irvine, CA 92697, USA

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     title = {On {Monge{\textendash}Kantorovich} problem in the plane},
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Shen, Yinfang; Zheng, Weian. On Monge–Kantorovich problem in the plane. Comptes Rendus. Mathématique, Tome 348 (2010) no. 5-6, pp. 267-271. doi : 10.1016/j.crma.2009.11.022. http://archive.numdam.org/articles/10.1016/j.crma.2009.11.022/

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