Optimal Transport in Competition with Reaction: The Hellinger--Kantorovich Distance and Geodesic Curves

We discuss a new notion of distance on the space of finite and nonnegative measures on $\Omega \subset {\mathbb R}^d$, which we call the Hellinger--Kantorovich distance. It can be seen as an inf-convolution of the well-known Kantorovich--Wasserstein distance and the Hellinger-Kakutani distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and some of its properties. In particular, the distance can be equivalently described by an optimal transport problem on the cone space over the underlying space $\Omega$. We give a construction of geodesic curves and discuss examples and their general properties