Stochastic differential equations and stochastic parallel translations in the Wasserstein space

We will develop some elements in stochastic analysis in the Wasserstein space ${\mathbb{P}}_2(M)$ over a compact Riemannian manifold $M$, such as intrinsic It$\widehat{o}$ formulae, stochastic regular curves and parallel translations along them. We will establish the existence of parallel translations along regular curves, or stochastic regular curves in case of ${\mathbb{P}}_2(\mathbb{T})$. Surprisingly enough, in this last case, the equation defining stochastic parallel translations is a SDE on a Hilbert space, instead of a SPDE.