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

TL;DR

This paper develops stochastic calculus on the Wasserstein space over a Riemannian manifold, establishing existence and uniqueness of solutions, and explores equivariant diffusions on the diffeomorphism group with a focus on geometric and probabilistic structures.

Contribution

It introduces a framework for stochastic differential equations on Wasserstein space and analyzes equivariant diffusions on the diffeomorphism group, connecting geometric structures with stochastic analysis.

Findings

Existence of solutions to stochastic differential equations on Wasserstein space.

Unique stochastic parallel transport along diffusions on Wasserstein space.

Factorization of equivariant diffusions into horizontal and vertical components.

Abstract

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 $\mathscr{D}$ as a Riemannian submersion onto the smooth Wasserstein space $\mathscr{P}_\infty$, we further prove the existence and uniqueness of the stochastic parallel parallel transport along diffusions on $\mathscr{P}_\infty$. Finally, we show that equivariant diffusions on $\mathscr{D}$ endowed with a principal bundle structure over $\mathscr{P}_\infty$ 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.