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

Aymeric Martin

TL;DR

This work develops a rigorous stochastic calculus on the Wasserstein space over a closed manifold by exploiting the group of diffeomorphisms as a Riemannian submersion. It proves existence/uniqueness of SDEs on the Wasserstein space, constructs a stochastic parallel transport along its diffusions, and shows a unique factorization of equivariant diffusions on the diffeomorphism group into a horizontal diffusion and a vertical right-exponential component. The analysis relies on an infinite-dimensional Riemannian-submersion geometry, an ILH principal-bundle structure, Nash embedding techniques, and a normal-tensor framework to couple horizontal and vertical dynamics. These results advance stochastic analysis in optimal-transport geometry and provide a foundation for stability studies of measure-valued diffusions and their tangent processes.

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 $¶_\infty$, we further prove the existence and uniqueness of the stochastic parallel parallel transport along diffusions on $¶_\infty$. Finally, we show that equivariant diffusions on $\mathscr{D}$ endowed with a principal bundle structure over $¶_\infty$ 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.

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

TL;DR

This work develops a rigorous stochastic calculus on the Wasserstein space over a closed manifold by exploiting the group of diffeomorphisms as a Riemannian submersion. It proves existence/uniqueness of SDEs on the Wasserstein space, constructs a stochastic parallel transport along its diffusions, and shows a unique factorization of equivariant diffusions on the diffeomorphism group into a horizontal diffusion and a vertical right-exponential component. The analysis relies on an infinite-dimensional Riemannian-submersion geometry, an ILH principal-bundle structure, Nash embedding techniques, and a normal-tensor framework to couple horizontal and vertical dynamics. These results advance stochastic analysis in optimal-transport geometry and provide a foundation for stability studies of measure-valued diffusions and their tangent processes.

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 as a Riemannian submersion onto the smooth Wasserstein space , we further prove the existence and uniqueness of the stochastic parallel parallel transport along diffusions on . Finally, we show that equivariant diffusions on endowed with a principal bundle structure over 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 of the group of volume preserving diffeomorphisms.

Paper Structure

This paper contains 24 sections, 65 theorems, 412 equations.

Table of Contents

  1. Introduction
  2. Finite-dimensional case
  3. Stochastic differential equations on Riemannian manifolds
  4. Stochastic parallel transport and Riemannian submersion
  5. Equivariant diffusions on principal bundles
  6. Principal bundle with a Riemannian submersion structure
  7. Infinite-dimensional geometric setting
  8. The Wasserstein space
  9. The group of diffeomorphisms
  10. $\mathscr{D}$ as a Riemannian submersion onto $\mathscr{P}_\infty$
  11. Infinite-dimensional stochastic calculus
  12. Stochastic differential equations on $\mathscr{D}$
  13. Stochastic differential equations on $\mathscr{P}$
  14. Preliminaries
  15. Proof of Theorem \ref{['Existence Unicité']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\bar{Z}_0, \bar{Z}_1, \dots, \bar{Z}_N$ be regular vector fields on $\mathscr{P}$. Then, the SDE admits a solution on $\mathscr{P}$ given by $\mu_t =(X_t)_\sharp \mu$ where $(X_t)_{t \geq 0}$ is the solution to a conditional McKean-Vlasov equation. Moreover, if $\mu \in \mathscr{P}_\infty$, $(\mu_t)_{t \geq 0}$ lies in $\mathscr{P}_\infty$.

Theorems & Definitions (189)

  • Theorem 1.1: Informal version of Theorem \ref{['Existence Unicité']}
  • Theorem 1.2: Informal version of Theorem \ref{['Existence unicité transport parallèle']}
  • Proposition 2.1
  • proof
  • Theorem 2.2: Stochastic Cauchy-Lipschitz
  • proof
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.2
  • ...and 179 more