A large deviation approach to optimal transport

2007/10/08 by Christian Léonard, Léonard, Christian · 2 citations
Mathematics · #49J45 #49J53 #58E99 #60F10 #60G57 #90B06 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Markov Chains and Monte Carlo Methods #Optimization and Control (math.OC) #Probability (math.PR) #math.OC #math.PR #msc:49J45 #msc:49J53 #msc:58E99 #msc:60F10 #msc:60G57 #msc:90B06

paper · pdf · doi:10.48550/arxiv.0710.1461

arxiv created 2007/10/08 · openalex publication_date 2007/10/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02

Abstract

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.

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