The Hellinger-Kantorovich metric measure geometry on spaces of measures
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2025
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| author | Schiavo, Lorenzo Dello Sodini, Giacomo Enrico |
| author_facet | Schiavo, Lorenzo Dello Sodini, Giacomo Enrico |
| contents | 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 $\mathcal{L}_θ$, $θ>0$, 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 $\mathcal{L}_θ$ is the unique invariant measure; and (b) the metric measure structure of $\left(\mathcal{M}(M),\mathsf{H\! K}_{\mathsf{d}_g},\mathcal{L}_θ\right)$, 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 $θ\in (0,1]$, and properly associated with the Brownian motion of the Hellinger-Kantorovich geometry on $\mathcal{M}(M)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_07802 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Hellinger-Kantorovich metric measure geometry on spaces of measures Schiavo, Lorenzo Dello Sodini, Giacomo Enrico Functional Analysis Metric Geometry Probability 31C25, 49Q22 (primary), 60J60, 60G55, 22E66 (secondary) 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 $\mathcal{L}_θ$, $θ>0$, 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 $\mathcal{L}_θ$ is the unique invariant measure; and (b) the metric measure structure of $\left(\mathcal{M}(M),\mathsf{H\! K}_{\mathsf{d}_g},\mathcal{L}_θ\right)$, 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 $θ\in (0,1]$, and properly associated with the Brownian motion of the Hellinger-Kantorovich geometry on $\mathcal{M}(M)$. |
| title | The Hellinger-Kantorovich metric measure geometry on spaces of measures |
| topic | Functional Analysis Metric Geometry Probability 31C25, 49Q22 (primary), 60J60, 60G55, 22E66 (secondary) |
| url | https://arxiv.org/abs/2503.07802 |