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

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 $\mathrm{d}\mathrm{o}\mathrm{m}(\mathcal{E})$ and that $W_2$ is dominated by the intrinsic metric induced by $\mathcal{E}$. We illustrate our results by giving several detailed examples.