On the smooth Lorentzian optimal transport problem

TL;DR

This paper extends key concepts of optimal transport theory from Riemannian to Lorentzian geometry, establishing duality, characterizing optimality, and analyzing regularity of potentials under new assumptions.

Contribution

It provides the first Lorentzian analogs of duality, optimality characterization, and regularity results, with specific conditions ensuring these properties.

Findings

Strong duality holds under certain measure assumptions.

Optimality characterized by $c$-cyclical monotonicity in Lorentzian setting.

Weak potentials are locally semiconvex under suitable conditions.

Abstract

In this paper, we want to establish some general results in the Lorentzian optimal transport theory that have well-known Riemannian counterparts. As a first result, we will provide non-trivial assumptions on the measures to ensure strong (Kantorovich) duality, i.e. the existence of a maximizing pair in the dual optimal transport formulation. We will also show that, under suitable assumptions, optimality of a causal coupling is characterized by $c$-cyclical monotonicity. The second result deals with the regularity of $c$-convex functions and (weak) Kantorovich potentials. We show that, in general, regularity results known in the Riemannian context do not extend to the Lorentzian setting. Under suitable assumptions on the measures, we will prove nevertheless that these (weak) potentials are locally semconvex on an open set of full measure. As our last result, we will prove that, under quite general assumptions, for any two intermediates measures along a displacement interpolation, there exists a $C^{1,1}_{loc}$-regular maximizing pair in the dual formulation.