A Rademacher-type theorem on L2-Wasserstein spaces over closed Riemannian manifolds

Let P be any Borel probability measure on the L-2-Wasserstein space (P-2 (M), W-2) over a closed Riemannian manifold M. We consider the Dirichlet form epsilon induced by P and by the Wasserstein gradient on P-2(M). Under natural assumptions on P, we show that W-2-Lipschitz functions on , P-2(M) are contained in the Dirichlet space D(epsilon) and that W-2 is dominated by the intrinsic metric induced by epsilon. We illustrate our results by giving several detailed examples. (C) 2019 Elsevier Inc. All rights reserved.