The tangent space to the Wasserstein space: parallel transport and other applications

We propose a new notion of the formal tangent space to the Wasserstein space P(X) at a given measure. Modulo an integrability condition, we say that this tangent space is made of functions over X which are valued in the probability measures over the tangent bundle to X. This generalization of previous concepts of tangent spaces allows us to define appropriate notions of parallel transport, C1,α regularity over P(X) and translation of a curve over P(X).