The Hellinger-Kantorovich metric measure geometry on spaces of measures

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\! Kdg induced by dg. Firstly, we prove that (M(M),H\! Kdg) 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_θ, θ>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_θ is the unique invariant measure; and (b) the metric measure structure of (M(M),H\! Kdg,Lθ), 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 θ∈ (0,1], and properly associated with the Brownian motion of the Hellinger-Kantorovich geometry on M(M).