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

TL;DR

This paper introduces the Hellinger-Kantorovich distance, a new measure on finite nonnegative measures combining transport and reaction, with a full characterization and geodesic construction.

Contribution

It presents a novel distance generalizing Kantorovich-Wasserstein by incorporating reaction dynamics, with a comprehensive characterization and geodesic curve construction.

Findings

Distance can be described via an optimal transport problem on a cone space.

Geodesic curves are explicitly constructed and analyzed.

The distance exhibits properties blending transport and reaction processes.

Abstract

We discuss a new notion of distance on the space of finite and nonnegative measures which can be seen as a generalization of the well-known Kantorovich-Wasserstein 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 its properties. In fact the distance can be equivalently described by an optimal transport problem on the cone space over the underlying metric space. We give a construction of geodesic curves and discuss their properties.