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$C_{loc}^{1,1}$ optimal pairs in the dual optimal transport problem for a Lorentzian cost along displacement interpolations

Alec Metsch

TL;DR

The paper develops a Lorentzian analogue of weak KAM-based regularity results for optimal transport by working with a cost $c_t$ derived from Lorentzian action on globally hyperbolic spacetimes. Through a careful blend of Lax–Oleinik semigroup techniques, calibrated curves, and displacement interpolations, it proves the existence of $C_{loc}^{1,1}$ optimal dual pairs for intermediate transport steps under the timelike-minimizer concentration condition. The main contribution is extending $C^{1,1}_{loc}$ regularity to dual potentials in the Lorentzian OT setting, enabling precise characterizations of optimality and regularity along displacement interpolations. The methods combine weak KAM ideas with Lorentzian geometry, offering a framework for regularity results in spacetime transport problems with non-Riemannian costs and potential implications for geometric analysis on spacetimes.

Abstract

We consider the optimal transportation problem on a globally hyperbolic spacetime with a cost function $c$, which corresponds to the optimal transportation problem on a complete Riemannian manifold where the cost function is given by the squared Riemannian distance. Building upon methods of weak KAM theory, we will establish the existence of $C_{loc}^{1,1}$ optimal pairs for the dual optimal transport problem for probability measures along displacement interpolations.

$C_{loc}^{1,1}$ optimal pairs in the dual optimal transport problem for a Lorentzian cost along displacement interpolations

TL;DR

The paper develops a Lorentzian analogue of weak KAM-based regularity results for optimal transport by working with a cost derived from Lorentzian action on globally hyperbolic spacetimes. Through a careful blend of Lax–Oleinik semigroup techniques, calibrated curves, and displacement interpolations, it proves the existence of optimal dual pairs for intermediate transport steps under the timelike-minimizer concentration condition. The main contribution is extending regularity to dual potentials in the Lorentzian OT setting, enabling precise characterizations of optimality and regularity along displacement interpolations. The methods combine weak KAM ideas with Lorentzian geometry, offering a framework for regularity results in spacetime transport problems with non-Riemannian costs and potential implications for geometric analysis on spacetimes.

Abstract

We consider the optimal transportation problem on a globally hyperbolic spacetime with a cost function , which corresponds to the optimal transportation problem on a complete Riemannian manifold where the cost function is given by the squared Riemannian distance. Building upon methods of weak KAM theory, we will establish the existence of optimal pairs for the dual optimal transport problem for probability measures along displacement interpolations.

Paper Structure

This paper contains 9 sections, 37 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. The Lagrangian
  3. Lax-Oleinik semigroup
  4. Lax-Oleinik evolution for $C^1$ Lipschitz functions
  5. Application to Optimal transport
  6. Appendix
  7. Semiconcave functions and boundedness in $C^2$
  8. Proof of Lemma \ref{['h1']}
  9. Review of Lorentzian geometry

Key Result

Theorem 1.1

Let $\mu_0,\mu_1\in {\cal P}(M)$ and assume that $C(\mu_0,\mu_1)$ is finite. Let $\pi \in \Gamma_o(\mu_0,\mu_1)$ and $\Pi$ be a dynamical optimal coupling associated with $\pi$. Fix $0<s<t<1$. Suppose that $(\varphi,\psi)$ is an optimal pair in z and that $\pi(I^+)=1$. Then, there exists an optimal such that $\Phi_s$ is $C_{loc}^{1,1}$ on an open set of full $\mu_s$-measure and $\Psi_t$ is $C_{lo

Theorems & Definitions (113)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof : Sketch of proof
  • ...and 103 more