Kinetic Optimal Transport

We present a kinetic version of the optimal transport problem for probability measures on phase space. The central object is a second-order discrepancy between probability measures, analogous to the 2-Wasserstein distance, but based on the minimisation of the squared acceleration. We discuss the equivalence of static and dynamical formulations and characterise absolutely continuous curves of measures in terms of reparametrised solutions to the Vlasov continuity equation. This is based on joint work with Giovanni Brigati (ISTA) and Filippo Quattrocchi (ISTA).