Stochastic parallel transport on the Wasserstein space and equivariant diffusions on the group of diffeomorphisms over a closed Riemannian manifold

In this work, we establish the existence of solutions to stochastic differential equations on the Wasserstein space over a closed Riemannian manifold, under suitable regularity assumptions on the driving vector fields. Interpreting the diffeomorphism group $\mathcal{D}$ as a Riemannian submersion onto the smooth Wasserstein space ${\mathcal{P}}_{\mathrm{\infty }}$, we further prove the existence and uniqueness of the stochastic parallel parallel transport along diffusions on ${\mathcal{P}}_{\mathrm{\infty }}$. Finally, we show that equivariant diffusions on $\mathcal{D}$ endowed with a principal bundle structure over ${\mathcal{P}}_{\mathrm{\infty }}$ admit a unique factorization into a horizontal diffusion and a vertical component expressed as a right exponential of a process taking values in the Lie algebra $\mathfrak{g}$ of the group $G$ of volume preserving diffeomorphisms.