On entropy-transport problems and the Hellinger--Kantorovich distance

In  this  talk, we  will  present  a general  class  of variational problems involving entropy-transport minimization with  respect to a couple of given finite measures with possibly unequal total mass. These optimal entropy-transport problems can be regarded as a  natural generalization of classical optimal transportation problems. With an appropriate choice of the entropy/cost functionals they provide a distance between measures that exhibits interesting geometric features.  We call this distance Hellinger-Kantorovich distance as it can be seen as an interpolation between the Hellinger and the Kantorovich-Wasserstein distance. The link to the entropy-
transport minimization problems relies on convex duality in a surprising way. Moreover, a dynamic Benamou-Brenier characterization also shows the role of these distances in dynamic processes involving creation or annihilation of masses. Finally, we will  give a characterization of geodesic curves and of convex functionals. It is a joint work with Giuseppe Savaré and Alexander Mielke.