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arXiv · 2609.10875

Metric Geometry of the Signature Group for $p$-Variation Rough Paths

Felix Medwed·Sylvie Paycha·Alexander Schmeding
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Abstract

The signatures of $p$-rough paths form a subgroup of sufficiently high-level truncated tensor algebras, whose inverse limit is a subgroup of the full tensor algebra. For $p \geq 1$, we provide a top-down description of the signature group as the inverse limit of finite-dimensional Carnot--Carathéodory geometries in the $p$-variation setting. We show that every compatible choice of metrics induces a topological tree structure on the inverse-limit group, under which the signature group is not a topological group. This extends the results of Enrico Le Donne and Roland Züst from bounded variation to rough paths. We also characterise the dependence of the inverse-limit groups and their metric completions on the choice of metric, identifying them with the tree-reduced path group of Horatio Boedihardjo, Xiang Geng, Terry Lyons, and Danyu Yang.

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Felix Medwed, Sylvie Paycha, Alexander Schmeding. 2026-09-09. Metric Geometry of the Signature Group for $p$-Variation Rough Paths. https://arxiv.org/abs/2609.10875

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