Stochastic differential equations and stochastic parallel translations in the Wasserstein space

TL;DR

This paper develops stochastic analysis tools like Itô formulae and parallel translations within the Wasserstein space over manifolds, revealing novel SDE formulations for stochastic parallel translations.

Contribution

It introduces stochastic analysis elements in Wasserstein space, including intrinsic Itô formulae and the existence of stochastic parallel translations, with a surprising SDE formulation in certain cases.

Findings

Existence of parallel translations along regular curves.

Development of intrinsic Itô formulae in Wasserstein space.

Stochastic parallel translation equations as SDEs on Hilbert spaces.

Abstract

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.