Limit Theorems for Optimal Mass Transportation

TL;DR

This paper explores new theoretical links in optimal mass transportation, revealing a surprising connection between different Lagrangian actions on Riemannian manifolds and providing a novel limit formula for Wasserstein distances.

Contribution

It introduces a new link between optimal transformations from different Lagrangian actions and establishes a limit relation for Wasserstein distances on Riemannian manifolds.

Findings

Established a limit formula for Wasserstein distances involving small perturbations.

Linked optimal mass transportation to Lagrangian actions on Riemannian manifolds.

Provided theoretical insights into the structure of optimal transport maps.

Abstract

The optimal mass transportation was introduced by Monge some 200 years ago and is, today, the source of large number of results in analysis, geometry and convexity. Here I investigate a new, surprising link between optimal transformations obtained by different Lagrangian actions on Riemannian manifolds. As a special case, for any pair of non-negative measures $\lambda^+,\lambda^-$ of equal mass $$ W_1(\lambda^-, \lambda^+)= \lim_{\eps\to 0} \eps^{-1}\inf_{\mu} W_p(\mu+\eps\lambda^-, \mu+\eps\lambda^+)$$ where $W_p$, $p\geq 1$ is the Wasserstein distance and the infimum is over the set of probability measures in the ambient space.