Entropic Selection Principle for Monge's Optimal Transport

Abstract

We investigate the small regularization limit of entropic optimal transport when the cost function is the Euclidean distance in dimensions $d>1$, and the marginal measures are absolutely continuous with respect to the Lebesgue measure. Our results establish that the limiting optimal transport plan is supported on transport rays. Furthermore, within each transport ray, the limiting transport plan uniquely minimizes a relative entropy functional with respect to specific reference measures supported on the rays. This provides a complete and unique characterization of the limiting transport plan. While similar results have been obtained for $d=1$ in \cite{Marino} and for discrete measures in \cite{peyré2020computationaloptimaltransport}, this work resolves the previously open case in higher dimensions $d>1.$