发表机构
University of Potsdam; Norwegian University of Science and Technology（波茨坦大学; 挪威科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在$p$-变差框架下，将签名群刻画为有限维Carnot--Carathéodory几何的逆极限，证明其非拓扑群，并揭示其度量完备化与树约化路径群的等价性。
AI 中文摘要
$p$-粗糙路径的签名构成足够高阶截断张量代数的子群，其逆极限是全张量代数的子群。对于 $p \geq 1$，我们在 $p$-变差框架下，将签名群描述为有限维Carnot--Carathéodory几何的逆极限，从而给出其自上而下的刻画。我们证明，每一个相容的度量选择都会在逆极限群上诱导出一种拓扑树结构，在此结构下，签名群不是拓扑群。这将对Enrico Le Donne和Roland Züst的结果从有界变差情形推广到粗糙路径情形。我们还刻画了逆极限群及其度量完备化对度量选择的依赖性，并将其与Horatio Boedihardjo、Xiang Geng、Terry Lyons和Danyu Yang提出的树约化路径群等同起来。
英文摘要
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.
Comments42 pages, 1 table
