On the existence of dual solutions for Lorentzian cost functions

TL;DR

This paper investigates the dual solutions in Lorentzian optimal transportation, establishing conditions for their existence, and demonstrates that dual solutions imply timelike transport, also addressing absolute continuity and the relativistic Monge problem.

Contribution

It provides new conditions under which dual solutions exist in Lorentzian optimal transport and connects dual solutions to timelike transport and absolute continuity.

Findings

Dual solutions do not always exist without assumptions.

Existence of dual solutions implies timelike transport on full measure.

Absolute continuity persists along optimal transport under certain conditions.

Abstract

The dual problem of optimal transportation in Lorentz-Finsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal transport is timelike on a set of full measure. In the second part the persistence of absolute continuity along an optimal transportation under obvious assumptions is proven and a solution to the relativistic Monge problem is provided.