The tangent space to the Wasserstein space: parallel transport and other applications
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909954965766144 |
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| author | Bertucci, Charles |
| author_facet | Bertucci, Charles |
| contents | We propose a new notion of the formal tangent space to the Wasserstein space $\mathcal{P}(X)$ at a given measure. Modulo an integrability condition, we say that this tangent space is made of functions over $X$ which are valued in the probability measures over the tangent bundle to $X$. This generalization of previous concepts of tangent spaces allows us to define appropriate notions of parallel transport, $\mathcal{C}^{1,α}$ regularity over $\mathcal{P}(X)$ and translation of a curve over $\mathcal{P}(X)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09763 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The tangent space to the Wasserstein space: parallel transport and other applications Bertucci, Charles Analysis of PDEs Metric Geometry Optimization and Control We propose a new notion of the formal tangent space to the Wasserstein space $\mathcal{P}(X)$ at a given measure. Modulo an integrability condition, we say that this tangent space is made of functions over $X$ which are valued in the probability measures over the tangent bundle to $X$. This generalization of previous concepts of tangent spaces allows us to define appropriate notions of parallel transport, $\mathcal{C}^{1,α}$ regularity over $\mathcal{P}(X)$ and translation of a curve over $\mathcal{P}(X)$. |
| title | The tangent space to the Wasserstein space: parallel transport and other applications |
| topic | Analysis of PDEs Metric Geometry Optimization and Control |
| url | https://arxiv.org/abs/2512.09763 |