Optimal transport and regularity of weak Kantorovich potentials on a globally hyperbolic spacetime

TL;DR

This paper studies the regularity of weak Kantorovich potentials in optimal transport on globally hyperbolic spacetimes, establishing existence, uniqueness, and structure of optimal maps under certain conditions.

Contribution

It extends optimal transport theory to globally hyperbolic spacetimes, analyzing the regularity of weak potentials and deriving conditions for optimal map existence and uniqueness.

Findings

Regularity results for weak Kantorovich potentials

Existence and uniqueness of optimal transport maps

Structural insights into optimal transport on spacetimes

Abstract

We consider the optimal transportation problem on a globally hyperbolic spacetime for some cost function $c_2$, which corresponds to the optimal transportation problem on a complete Riemannian manifold where the cost function is the Riemannian distance squared. Building on insights from previous studies on the Riemannian and Lorentzian case, our main goal is to investigate the regularity of $\pi$-solutions (weak versions of Kantorovich potentials), from which we can conclude, in a classical way, the existence, uniqueness and structure of an optimal transport map between given Borel probability measures $\mu$ and $\nu$, under suitable assumptions.