A Rademacher-type Theorem on $L^2$-Wasserstein Spaces over Closed Riemannian Manifolds

Abstract

Let $\mathbb{P}$ be any Borel probability measure on the $L^2$-Wasserstein space $({\mathcal{P}}_2(M),W_2)$ over a closed Riemannian manifold $M$. We consider the Dirichlet form $\mathcal{E}$ induced by $\mathbb{P}$ and by the Wasserstein gradient on ${\mathcal{P}}_2(M)$. Under natural assumptions on $\mathbb{P}$, we show that $W_2$-Lipschitz functions on ${\mathcal{P}}_2(M)$ are contained in the Dirichlet space $\mathrm{dom}(\mathcal{E})$ and that $W_2$ is dominated by the intrinsic metric induced by $\mathcal{E}$. We illustrate our results by giving several detailed examples.

arXiv comment40 pages; to appear in J. Funct. Anal. 278(6) 2020

References in corpus (4)

• The Dirichlet-Ferguson Diffusion on the Space of Probability Measures over a Closed Riemannian Manifold
• A Lack of Ricci Bounds for the Entropic Measure on Wasserstein space over the Interval
• Characteristic Functionals of Dirichlet Measures
• A Mecke-type characterization of the Dirichlet-Ferguson measure

Cited by in corpus (5)

• The Dirichlet-Ferguson Diffusion on the Space of Probability Measures over a Closed Riemannian Manifold
• Dissipative probability vector fields and generation of evolution semigroups in Wasserstein spaces
• Rademacher-type Theorems and Sobolev-to-Lipschitz Properties for Strongly Local Dirichlet Spaces
• Configuration spaces over singular spaces -- I. Dirichlet-Form and Metric Measure Geometry
• Sobolev-to-Lipschitz Property on QCD-spaces and Applications