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Optimal transport on the sub-Lorentzian Heisenberg group

Samuël Borza, Wilhelm Klingenberg, Patrick Wood

TL;DR

This work extends Lorentzian optimal transport theory to the sub-Lorentzian Heisenberg group by formulating the forward (and backward) Lorentz–Monge problem with cost $c_p= au^p/p$ for $p\in(0,1)$. It proves a sub-Lorentzian Brenier theorem, showing the optimal transport map is unique and expressible via a sub-Lorentzian exponential driven by a $c_p$-concave potential, with a companion Monge–Ampère type equation relating push-forward densities. The authors develop a robust Lagrange multiplier/Pontryagin framework adapted to timelike covectors, establish duality and $c_p$-cyclic monotonicity, and derive regularity results for the time-separation function that underlie the theory. They also present concrete transport map constructions and a Monge–Ampère-type PDE, linking geometric control, non-smooth Lorentzian geometry, and transport in low-regularity spacetimes. The results advance synthetic Lorentzian geometry on sub-Riemannian-like spaces and provide tools for studying transport problems in non-smooth spacetime models with explicit geometric maps and PDE characterizations.

Abstract

We investigate the synthetic metric spacetime structure of the sub-Lorentzian Heisenberg group and we study the optimal transport problem in this space. The sub-Lorentzian version of Brenier's theorem is established in this setting. Finally, we provide examples of optimal transport maps and derive a sub-Lorentzian Monge-Ampère equation.

Optimal transport on the sub-Lorentzian Heisenberg group

TL;DR

This work extends Lorentzian optimal transport theory to the sub-Lorentzian Heisenberg group by formulating the forward (and backward) Lorentz–Monge problem with cost for . It proves a sub-Lorentzian Brenier theorem, showing the optimal transport map is unique and expressible via a sub-Lorentzian exponential driven by a -concave potential, with a companion Monge–Ampère type equation relating push-forward densities. The authors develop a robust Lagrange multiplier/Pontryagin framework adapted to timelike covectors, establish duality and -cyclic monotonicity, and derive regularity results for the time-separation function that underlie the theory. They also present concrete transport map constructions and a Monge–Ampère-type PDE, linking geometric control, non-smooth Lorentzian geometry, and transport in low-regularity spacetimes. The results advance synthetic Lorentzian geometry on sub-Riemannian-like spaces and provide tools for studying transport problems in non-smooth spacetime models with explicit geometric maps and PDE characterizations.

Abstract

We investigate the synthetic metric spacetime structure of the sub-Lorentzian Heisenberg group and we study the optimal transport problem in this space. The sub-Lorentzian version of Brenier's theorem is established in this setting. Finally, we provide examples of optimal transport maps and derive a sub-Lorentzian Monge-Ampère equation.

Paper Structure

This paper contains 9 sections, 32 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. The sub-Lorentzian Heisenberg group
  3. Spacetime geometry of the sub-Lorentzian Heisenberg group
  4. The Lagrange multiplier rule
  5. The Lorentzian metric spacetime structure
  6. Optimal Transport in the sub-Lorentzian Heisenberg group
  7. Lorentzian optimal transport in the Heisenberg group
  8. Brenier's theorem in the sub-Lorentzian Heisenberg group
  9. A sub-Lorentzian Monge-Ampère type equation and some examples

Key Result

Theorem 1

Let $p \in \ointerval{0}{1}$, $\mu, \nu \in \mathcal{P}(\mathds{H})$ with compact support, $\mu, \nu \ll \mathcal{L}^3$, and $\mathrm{supp}(\mu) \times \mathrm{supp}(\nu) \subseteq \mathds{H}_{\ll}^2$. Then, the following holds.

Theorems & Definitions (66)

  • Theorem 1: Brenier's theorem in the sub-Lorentzian Heisenberg group
  • Definition 2
  • Theorem 3: Pontryagin's Maximum Principle
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Definition 7
  • ...and 56 more