Let (M, g) be a Riemannian manifold with Riemannian distance dg, and M (M) be the space of all non-negative Borel measures on M, endowed with the Hellinger-Kantorovich distance H\! K₃₆ induced by dg. Firstly, we prove that (M (M), H\! K₃₆) is a universally infinitesimally Hilbertian metric space, and that a natural class of cylinder functions is dense in energy in the Sobolev space of every finite Borel measure on M (M). Secondly, we endow M (M) with its canonical reference measure, namely A. M. Vershik's multiplicative infinite-dimensional Lebesgue measure L_, >0, and we consider: (a) the geometric structure on M (M) induced by the natural action on M (M) of the semi-direct product of diffeomorphisms and densities on M, under which L_ is the unique invariant measure; and (b) the metric measure structure of (M (M), H\! K₃₆, L_), inherited from that of (M, dg, volg). We identify the canonical Dirichlet form (E, D (E) ) of (a) with the Cheeger energy of (b), thus proving that these two structures coincide. We further prove that (E, D (E) ) is a conservative quasi-regular strongly local Dirichlet form on M (M), recurrent if and only if (0, 1], and properly associated with the Brownian motion of the Hellinger-Kantorovich geometry on M (M).
Schiavo et al. (2025) studied this question.