WIAS Preprint No. 2926, (2022)

The geometry of controlled rough paths


  • Varzaneh, Mazyar Ghani
  • Riedel, Sebastian
  • Schmeding, Alexander
  • Tapia, Nikolas
    ORCID: 0000-0003-0018-2492

2020 Mathematics Subject Classification

  • 34K50 37H10 37H15


  • Continuous fields of Banach spaces,, rough paths, controlled rough paths




We prove that the spaces of controlled (branched) rough paths of arbitrary order form a continuous field of Banach spaces. This structure has many similarities to an (infinite-dimensional) vector bundle and allows to define a topology on the total space, the collection of all controlled path spaces, which turns out to be Polish in the geometric case. The construction is intrinsic and based on a new approximation result for controlled rough paths. This framework turns well-known maps such as the rough integration map and the Itô-Lyons map into continuous (structure preserving) mappings. Moreover, it is compatible with previous constructions of interest in the stability theory for rough integration.

Download Documents