Appears in collection : Journée sous-riemannienne 2018

This presentation is devoted to the study of mass transportation on sub-Riemannian geometry. In order to obtain existence and uniqueness of optimal transport maps, the first relevant method to consider is the one used by Figalli and Rifford which is based on the local semiconcavity of the sub-Riemannian distance outside the diagonal. Recently, Cavalletti and Huesmann developed a new method to solve the Monge problem using a measure contraction property. That is why we attempt to prove the MCP property on sub-Riemannian structures as a consequence of the upper bound of the horizontal Hessian of the sub-Riemannian distance.