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arXiv · 1810.10227

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

Lorenzo Dello Schiavo
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Abstract

Let $\mathbb P$ be any Borel probability measure on the $L^2$-Wasserstein space $(\mathscr{P}_2(M),W_2)$ over a closed Riemannian manifold $M$. We consider the Dirichlet form $\mathcal E$ induced by $\mathbb P$ and by the Wasserstein gradient on $\mathscr{P}_2(M)$. Under natural assumptions on $\mathbb P$, we show that $W_2$-Lipschitz functions on $\mathscr{P}_2(M)$ 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.

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BibTeXRIS

Lorenzo Dello Schiavo. 2018-10-24. A Rademacher-type Theorem on $L^2$-Wasserstein Spaces over Closed Riemannian Manifolds. https://doi.org/10.1016/j.jfa.2019.108397

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