PDE aspects of the dynamical optimal transport in the Lorentzian setting

TL;DR

This paper extends the connection between optimal transport and PDEs from Riemannian manifolds to Lorentzian spacetimes, introducing a causal continuity inequality and a Lorentzian Benamou--Brenier formula.

Contribution

It introduces a Lorentzian analogue of the Benamou--Brenier formula and establishes a causal continuity inequality linking PDEs and optimal transport in Lorentzian geometry.

Findings

Established a Lorentzian version of the Benamou--Brenier formula.

Defined a causal continuity inequality for spacetimes.

Linked PDEs with optimal transport in Lorentzian geometry.

Abstract

One of the crucial features of optimal transport on Riemannian manifolds is the equivalence of the `static', original, formulation of the problem and of the `dynamic' one, based on the study of the continuity equation. This furnishes the key link between Wasserstein geometry and PDEs that has found so many applications in the last 20 years. In this paper we investigate this kind of equivalence on spacetimes. At the PDE level, this requires to transition from the continuity equation to a suitable `continuity inequality', to which we shall refer to as `causal continuity inequality'. As a direct consequence of our findings we obtain a Lorentzian version of the celebrated Benamou--Brenier formula.