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arXiv · 2512.09763

The tangent space to the Wasserstein space: parallel transport and other applications

Charles Bertucci
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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, $\mathcal{C}^{1,\alpha}$ regularity over $\mathcal{P}(X)$ and translation of a curve over $\mathcal{P}(X)$.

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Charles Bertucci. 2025-12-10. The tangent space to the Wasserstein space: parallel transport and other applications. https://arxiv.org/abs/2512.09763

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