A Note on the Differentiability of the Hellinger–Kantorovich Distances

Abstract

This paper will deal with differentiability properties of the class of Hellinger–Kantorovich distances textsf{HK}_{Lambda , Sigma } (Lambda , Sigma > 0) which was recently introduced on the space {mathcal {M}}({mathbb {R}}^d) of finite nonnegative Radon measures. The derivatives of tmapsto textsf{HK}_{Lambda , Sigma }(mu _t, nu _t)^2, for absolutely continuous curves (mu _t)_t, (nu _t)_t in ({mathcal {M}}({mathbb {R}}^d),textsf{HK}_{Lambda , Sigma }), will be computed {mathscr {L}}^1-a.e.. The characterization of absolutely continuous curves in ({mathcal {M}}({mathbb {R}}^d), textsf{HK}_{Lambda , Sigma }) will be refined.