The Hellinger-Kantorovich metric measure geometry on spaces of measures

Abstract

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 ${\mathsf{H}\mathsf{K}}_{{\mathsf{d}}_g}$ induced by ${\mathsf{d}}_g$. Firstly, we prove that $\left(\mathcal{M}(M),{\mathsf{H}\mathsf{K}}_{{\mathsf{d}}_g}\right)$ 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 $\left(\mathcal{M}(M),{\mathsf{H}\mathsf{K}}_{{\mathsf{d}}_g},{\mathcal{L}}_{\theta }\right)$, inherited from that of $(M,{\mathsf{d}}_g,{\mathrm{vol}}_g)$. We identify the canonical Dirichlet form $\left(\mathcal{E},\mathcal{D}(\mathcal{E})\right)$ of (a) with the Cheeger energy of (b), thus proving that these two structures coincide. We further prove that $\left(\mathcal{E},\mathcal{D}(\mathcal{E})\right)$ 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 $\mathcal{M}(M)$.

96 pages