Limit theorems for optimal mass transportation

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 λ+, λ− of equal mass $$W_1(\lambda^-, \lambda^+)= \lim_{\varepsilon\rightarrow 0} \varepsilon^{-1} \inf_{\mu} W_p(\mu+\varepsilon\lambda^-, \mu+\varepsilon\lambda^+) $$ where Wp, p ≥ 1 is the Wasserstein distance and the infimum is over the set of probability measures in the ambient space.