The Hellinger-Kantorovich metric measure geometry on spaces of measures

Let $(M,g)$ be a Riemannian manifold with Riemannian distance $\mathsf{d}_g$, and $\mathcal{M}(M)$ be the space of all non-negative Borel measures on $M$, endowed with the Hellinger-Kantorovich distance $\mathsf{H\! K}_{\mathsf{d}_g}$ induced by $\mathsf{d}_g$. Firstly, we prove that $\left(\mathcal{M}(M),\mathsf{H\! K}_{\mathsf{d}_g}\right)$ 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 $\mathcal{M}(M)$. Secondly, we endow $\mathcal{M}(M)$ with its canonical reference measure, namely A.M. Vershik's multiplicative infinite-dimensional Lebesgue measure $\mathcal{L}_\theta$, $\theta>0$, and we consider: (a) the geometric structure on $\mathcal{M}(M)$ induced by the natural action on $\mathcal{M}(M)$ of the semi-direct product of diffeomorphisms and densities on $M$, under which $\mathcal{L}_\theta$ is the unique invariant measure; and (b) the metric measure structure of $\left(\mathcal{M}(M),\mathsf{H\! K}_{\mathsf{d}_g},\mathcal{L}_{\theta}\right)$, inherited from that of $(M,\mathsf{d}_g,\mathrm{vol}_g)$. We identify the canonical Dirichlet form $\left(\mathcal{E},\mathscr{D}(\mathcal{E})\right)$ of (a) with the Cheeger energy of (b), thus proving that these two structures coincide. We further prove that $\left(\mathcal{E},\mathscr{D}(\mathcal{E})\right)$ is a conservative quasi-regular strongly local Dirichlet form on $\mathcal{M}(M)$, recurrent if and only if $\theta\in (0,1]$, and properly associated with the Brownian motion of the Hellinger-Kantorovich geometry on $\mathcal{M}(M)$.