Some notes on the Hellinger distance and various Fisher-Rao distances

These expository notes introduce the Hellinger distance on the set of all measures and the induced Fisher-Rao distances for subsets of measures, such as probability measures or Gaussian measures. The historical background is highlighted and the relations and the distinct features of the two distances are discussed. Moreover, we provide a dynamic characterization of absolutely continuous curves in the Hellinger spaces in terms of the growth equation, which replaces the continuity equation in the theory of optimal transport.