Felix Medwed (Universität Potsdam)
E. Le Donne and R. Züst (2021) showed that the space of signatures of rectifiable paths is a geodesic metric tree. We extend their result originally formulated in terms of curve length to the setting of $p$-variation for $p>1$. This extension relies on results of Boedihardjo, Geng, Lyons, and Yang (2016) concerning signatures of weakly geometric rough paths, which provide the appropriate analytical control in the $p$-variation framework. The key components of the original construction are reinterpreted using the language of metric groups, allowing us to define and work with the signature group adapted to $p$-variation. Within this framework, we revisit the lifting procedure introduced by Le Donne and Züst—namely, the construction of a canonical lift of a path into a suitable group structure—and show that the main result, concerning the existence and properties of such lifts, continues to hold when length is replaced by $p$-variation.
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