Finding smooth interpolations between probability measures is a problem of broad interest, with natural applications, e.g., in biology (trajectory inference) and computer graphics (image interpolation). In this talk, I will discuss a model in which such interpolations are obtained by minimizing an action functional of the acceleration. This minimization defines a discrepancy between measures that -- in analogy with Wasserstein distances from optimal transport theory -- admits an equivalent fluid-dynamical formulation and induces a Riemannian-like geometry on the space of measures. 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 University-PSL).
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006