Transport densities and congested optimal transport problem in the Heisenberg group

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.