An intrinsic parallel transport in Wasserstein space

Abstract

If M M is a smooth compact connected Riemannian manifold, let P ( M ) P(M) denote the Wasserstein space of probability measures on M M . We describe a geometric construction of parallel transport of some tangent cones along geodesics in P ( M ) P(M) . We show that when everything is smooth, the geometric parallel transport agrees with earlier formal calculations.