The study demonstrates the metric measure structure of measures on Riemannian manifolds, suggesting deep geometric properties.
Key Points
The space of measures on a Riemannian manifold is universally infinitesimally Hilbertian, supporting a coherent geometric framework.
The canonical Dirichlet form is identified with Cheeger energy, establishing a significant correspondence in metric measure theory.
The induced Hellinger-Kantorovich distance provides a unique invariant measure, enhancing the understanding of geometric structures in spaces of measures.
The study reveals that the quasi-regular Dirichlet form is recurrent under certain conditions, linking it to the behavior of Brownian motion on these measure spaces.