A large deviation approach to optimal transport

Christian Léonard (1, 2)

1 MODAL'X - Modélisation aléatoire de Paris X
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

A probabilistic method for solving the Monge-Kantorovich mass transport problem on $R^d$ is introduced. A system of empirical measures of independent particles is built in such a way that it obeys a doubly indexed large deviation principle with an optimal transport cost as its rate function. As a consequence, new approximation results for the optimal cost function and the optimal transport plans are derived. They follow from the Gamma-convergence of a sequence of normalized relative entropies toward the optimal transport cost. A wide class of cost functions including the standard power cost functions $|x-y|^p$ enter this framework.

Mots clés: Monge-Kantorovich mass transport problem; large deviations of empirical measures; Gamma-convergence; doubly indexed large deviation principles; relative entropy

Domaines: Probabilités [math.PR]; Optimisation et contrôle [math.OC]

Christian Léonard. A large deviation approach to optimal transport. 2007. ⟨hal-00177365⟩