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On the smooth Lorentzian optimal transport problem

Alec Metsch

TL;DR

The paper develops a Lorentzian optimal transport theory with cost $c(x,y)=-d(x,y)^p$ $(0<p<1)$, addressing duality, causal-monotonicity, and regularity of $c$-convex potentials under suitable measure conditions. It proves existence of a dual maximizing pair and strong duality using a Kell-type approach, and shows optimal couplings are concentrated on causal sets and are characterized by a gradient equation, yielding a map when regularity holds. It further demonstrates that Lorentzian geometry can break Riemannian regularity intuitions, providing counterexamples, yet establishes a Lorentzian weak KAM framework that yields $C^{1,1}_{loc}$ regularity for calibrated pairs along intermediate measures and along displacement interpolations. The results collectively furnish a foundational, regularity-aware Lorentzian OT theory with gradient-based characterizations and a path toward applications in Lorentzian geometry and relativity. The work thus extends duality, monotonicity, and regularity techniques to the Lorentzian setting, facilitating further geometric and physical analyses of causal transport.

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.

On the smooth Lorentzian optimal transport problem

TL;DR

The paper develops a Lorentzian optimal transport theory with cost , addressing duality, causal-monotonicity, and regularity of -convex potentials under suitable measure conditions. It proves existence of a dual maximizing pair and strong duality using a Kell-type approach, and shows optimal couplings are concentrated on causal sets and are characterized by a gradient equation, yielding a map when regularity holds. It further demonstrates that Lorentzian geometry can break Riemannian regularity intuitions, providing counterexamples, yet establishes a Lorentzian weak KAM framework that yields regularity for calibrated pairs along intermediate measures and along displacement interpolations. The results collectively furnish a foundational, regularity-aware Lorentzian OT theory with gradient-based characterizations and a path toward applications in Lorentzian geometry and relativity. The work thus extends duality, monotonicity, and regularity techniques to the Lorentzian setting, facilitating further geometric and physical analyses of causal transport.

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 -cyclical monotonicity. The second result deals with the regularity of -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 -regular maximizing pair in the dual formulation.

Paper Structure

This paper contains 14 sections, 47 theorems, 260 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Lagrangian
  3. Some optimal transport theory – Proofs of Theorem \ref{['thmb']} and Corollary \ref{['cor4']}
  4. Riemann vs Lorentz – Counterexamples
  5. Regularity of $c$-convex functions – Proof of Theorem \ref{['thmc']}
  6. Proof of Theorem \ref{['thmai']}
  7. Proof of Corollary \ref{['cor2']}
  8. A Lorentzian weak KAM result – Proof of Theorem \ref{['thmg']}
  9. Local semiconcavity
  10. Local semiconvexity
  11. Local semiconcavity again
  12. On the Lagrangian
  13. Construction of the $\pi$-solution
  14. Local semiconcavity

Key Result

Theorem 1.4

Let $\mu,\nu\in {\cal P}$.

Figures (4)

  • Figure 1: Construction of a $c$-convex function that is not continuous in the interior of its domain.
  • Figure 2: Construction of a $c$-convex function whose $c$-subdifferential is locally unbounded.
  • Figure 3: Vizualisation of an optimal coupling that is not supported on the set of chronologically related points.
  • Figure 4: Causal compactness is crucial

Theorems & Definitions (133)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Definition 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 123 more