Transport densities and congested optimal transport problem in the Heisenberg group

TL;DR

This work develops a continuous congested optimal transport theory on the Heisenberg group $\mathbb{H}^n$ by restricting transport to horizontal curves and formulating a convex minimization over horizontal traffic plans. It introduces horizontal transport density $a_\gamma$ and horizontal traffic intensity $i_Q$, derives absolute continuity and $L^p$ bounds under regularity assumptions, and defines a weighted distance $c_\phi$ for densities in $L^q$ with $q>N$. A convex congested OT problem is posed via $\min_Q \int_\Omega G(i_Q(x))\,dx$, and Wardrop equilibria are shown to correspond to equilibria of this congested system, giving existence results and links to a Monge–Kantorovich problem with a self-consistent congested metric. The results extend the Euclidean congested OT theory to a non-Euclidean, sub-Riemannian setting, enabling dual formulations and regularity investigations in $\mathbb{H}^n$.

Abstract

We adapt the problem of continuous congested optimal transport to the Heisenberg group, equipped with a sub-Riemannian metric. Originally introduced in the Euclidean setting by Carlier, Jimenez, and Santambrogio as a path-dependent variant of the Monge-Kantorovich problem, we significantly restrict the set of admissible curves to horizontal ones. We establish the existence of equilibrium configurations as solutions to a convex minimization problem over a suitable set of measures on horizontal curves. This result is achieved through the notions of horizontal transport density and horizontal traffic intensity.

Theorems & Definitions (40)

• Theorem 2.1
• Proposition 2.2
• Theorem 2.3
• Proposition 2.4
• Lemma 2.5
• Theorem 2.6
• Lemma 3.1
• proof
• Proposition 3.2
• proof