From the Schr\"odinger problem to the Monge-Kantorovich problem

TL;DR

This paper demonstrates that the Monge-Kantorovich optimal transport problem can be derived as the limit of entropy minimization problems when the fluctuation parameter approaches zero, establishing convergence of entropic solutions to optimal plans.

Contribution

It shows the convergence of entropy minimization to optimal transport, introduces new Gamma-convergence results, and explores dynamic-static problem connections.

Findings

Entropic values converge to the optimal transport cost as fluctuations decrease.

Limit points of entropic minimizers are optimal transport plans.

Established new Gamma-convergence results for these problems.

Abstract

The aim of this article is to show that the Monge-Kantorovich problem is the limit of a sequence of entropy minimization problems when a fluctuation parameter tends down to zero. We prove the convergence of the entropic values to the optimal transport cost as the fluctuations decrease to zero, and we also show that the limit points of the entropic minimizers are optimal transport plans. We investigate the dynamic versions of these problems by considering random paths and describe the connections between the dynamic and static problems. The proofs are essentially based on convex and functional analysis. We also need specific properties of Gamma-convergence which we didn't find in the literature. Hence we prove these Gamma-convergence results which are interesting in their own right.