{"ID":2830770,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.09763","arxiv_id":"2512.09763","title":"The tangent space to the Wasserstein space: parallel transport and other applications","abstract":"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 $\\mathcal{P}(X)$.","short_abstract":"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 c...","url_abs":"https://arxiv.org/abs/2512.09763","url_pdf":"https://arxiv.org/pdf/2512.09763v1","authors":"[\"Charles Bertucci\"]","published":"2025-12-10T15:45:53Z","proceeding":"math.AP","tasks":"[\"math.AP\",\"math.MG\",\"math.OC\"]","methods":"[]","has_code":false}