We present three different notions of optimal mass transportation between probability measures, with respect to some “total minimization of the acceleration”.
Our main result is a hierarchy relating all three definitions, where our techniques combine probability, kinetic equations, and optimal transport. Motivated by this, we introduce a new “kinetic optimal transport” discrepancy and we study the geometric properties induced by it on the space of probability measures. The content of the talk is taken from the recent preprint arxiv.org/abs/2502.15665 by Jan Maas, Filippo Quattrocchi (ISTA), and the speaker.