The geometry of the space of branched rough paths

Abstract

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 these 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.