The geometry of the space of branched rough paths
- Tapia, Nikolas
- Zambotti, Lorenzo
2010 Mathematics Subject Classification
- 60H10 16T05
- Rough paths, Hopf algebras, renormalization
We construct an explicit transitive free action of a Banach space of Hölder functions on the space of branched rough paths, which yields in particular a bijection between theses two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker-Campbell-Hausdorff formula, on a constructive version of the Lyons-Victoir extension theorem and on the Hairer-Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths.
- Proc. London Math. Soc. (3), 121 (2020), pp. 220--251, DOI 10.1112/plms.12311 .