Kinetic Optimal Transport

Abstract: Wasserstein distances can be characterized as the minimum of certain functionals of the speed along trajectories connecting two mass configurations. In this talk, I will introduce a discrepancy between measures on a phase space that is instead defined via a functional of the acceleration. This can be seen as a smooth interpolation problem, with natural applications, e.g., in biology (trajectory inference) and computer graphics (image interpolation). Although the acceleration-based discrepancy is not a genuine distance, it admits a fluid-dynamical formulation akin to the Benamou--Brenier formula and induces a Riemannian-like geometry on the space of measures on the phase space. These results suggest possible applications to kinetic PDEs. This talk is based on arXiv:2502.15665, in collaboration with G. Brigati (ISTA) and J. Maas (ISTA), and ongoing work with G. Brigati (ISTA), G. Carlier (CEREMADE, Paris Dauphine-PSL), and J. Dolbeault (CEREMADE, Paris Dauphine-PSL).