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We discuss a new notion of distance on the space of finite and nonnegativemeasures which can be seen as a generalization of the well-knownKantorovich-Wasserstein distance. The new distance is based on a dynamicalformulation given by an Onsager operator that is the sum of a Wassersteindiffusion part and an additional reaction part describing the generation andabsorption of mass.  We present a full characterization of the distance and its properties. Infact the distance can be equivalently described by an optimal transport problemon the cone space over the underlying metric space. We give a construction ofgeodesic curves and discuss their properties.

Optimal transport in competition with reaction: the Hellinger-Kantorovich distance and geodesic curves