  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$\^o$ 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.
