The geometry of the space of branched Rough Paths

TL;DR

This paper develops a mathematical framework that characterizes the space of branched rough paths as a principal homogeneous space over a Banach space, enabling a deeper understanding of its structure and automorphisms.

Contribution

It introduces an explicit transitive free action of a Banach space on branched rough paths, connecting them via the Baker-Campbell-Hausdorff formula and the Hairer-Kelly map.

Findings

Establishes a bijection between branched rough paths and a Banach space of H"older functions.

Provides a characterization of automorphisms of the space of branched rough paths.

Constructs a principal homogeneous space structure for branched rough paths.

Abstract

We construct an explicit transitive free action of a Banach space of H\"older 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.