Stochastic parallel transport on the Wasserstein space and equivariant diffusions on the group of diffeomorphisms over a closed Riemannian manifold
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918359873880064 |
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| author | Martin, Aymeric |
| author_facet | Martin, Aymeric |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02975 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic parallel transport on the Wasserstein space and equivariant diffusions on the group of diffeomorphisms over a closed Riemannian manifold Martin, Aymeric Probability Differential Geometry 60D05 (Primary), 60G44, 53C30, 49Q22, 22E65 (Secondary) 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. |
| title | Stochastic parallel transport on the Wasserstein space and equivariant diffusions on the group of diffeomorphisms over a closed Riemannian manifold |
| topic | Probability Differential Geometry 60D05 (Primary), 60G44, 53C30, 49Q22, 22E65 (Secondary) |
| url | https://arxiv.org/abs/2512.02975 |