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

TL;DR

This paper proves a Rademacher-type theorem for Lipschitz functions on the Wasserstein space over a closed Riemannian manifold, linking Lipschitz continuity to the Dirichlet form and intrinsic metric.

Contribution

It establishes that $W_2$-Lipschitz functions belong to the Dirichlet space and that the Wasserstein metric is dominated by the intrinsic metric induced by the Dirichlet form.

Findings

$W_2$-Lipschitz functions are in the Dirichlet space

The Wasserstein metric is dominated by the intrinsic metric

Provides detailed examples illustrating the results

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.