WIAS Preprint No. 2645, (2019)

The geometry of the space of branched rough paths


  • Tapia, Nikolas
    ORCID: 0000-0003-0018-2492
  • 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.

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