  This article presents a new class of distances between arbitrary nonnegative
Radon measures inspired by optimal transport. These distances are defined by
two equivalent alternative formulations: (i) a dynamic formulation defining the
distance as a geodesic distance over the space of measures (ii) a static
"Kantorovich" formulation where the distance is the minimum of an optimization
problem over pairs of couplings describing the transfer (transport, creation
and destruction) of mass between two measures. Both formulations are convex
optimization problems, and the ability to switch from one to the other
depending on the targeted application is a crucial property of our models. Of
particular interest is the Wasserstein-Fisher-Rao metric recently introduced
independently by Chizat et al. and Kondratyev et al. Defined initially through
a dynamic formulation, it belongs to this class of metrics and hence
automatically benefits from a static Kantorovich formulation.
