Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-Order Discrepancies Between Probability Measures

Abstract

This is the first part of a general description in terms of mass transport for time-evolving interacting particles systems, at a mesoscopic level. Beyond kinetic theory, our framework naturally applies in biology, computer vision, and engineering. The central object of our study is a new discrepancy $\mathsf d$ between two probability distributions in position and velocity states, which is reminiscent of the $2$-Wasserstein distance, but of second-order nature. We construct $\mathsf d$ in two steps. First, we optimise over transport plans. The cost function is given by the minimal acceleration between two coupled states on a fixed time horizon $T$. Second, we further optimise over the time horizon $T>0$. We prove the existence of optimal transport plans and maps, and study two time-continuous characterisations of $\mathsf d$. One is given in terms of dynamical transport plans. The other one -- in the spirit of the Benamou--Brenier formula -- is formulated as the minimisation of an action of the acceleration field, constrained by Vlasov's equations. Equivalence of static and dynamical formulations of $\mathsf d$ holds true. While part of this result can be derived from recent, parallel developments in optimal control between measures, we give an original proof relying on two new ingredients: Galilean regularisation of Vlasov's equations and a kinetic Monge--Mather shortening principle. Finally, we establish a first-order differential calculus in the geometry induced by $\mathsf d$, and identify solutions to Vlasov's equations with curves of measures satisfying a certain $\mathsf d$-absolute continuity condition. One consequence is an explicit formula for the $\mathsf d$-derivative of such curves.