An isomorphism between branched and geometric rough paths

TL;DR

This paper establishes a natural isomorphism between branched and geometric rough paths, generalizing previous results and providing tools for analyzing signatures and uniqueness in rough path theory.

Contribution

It introduces an explicit isomorphism between branched and geometric rough paths, extending prior work and enabling new analysis of signatures and probabilistic properties.

Findings

Signature of a branched rough path is trivial iff the path is tree-like.

Constructs a non-commutative Fourier transform for signatures.

Provides conditions for a random signature to be uniquely determined by its expectation.

Abstract

We exhibit an explicit natural isomorphism between spaces of branched and geometric rough paths. This provides a multi-level generalisation of the isomorphism of Lejay-Victoir (2006) as well as a canonical version of the It\^o-Stratonovich correction formula of Hairer-Kelly (2015). Our construction is elementary and uses the property that the Grossman-Larson algebra is isomorphic to a tensor algebra. We apply this isomorphism to study signatures of branched rough paths. Namely, we show that the signature of a branched rough path is trivial if and only if the path is tree-like, and construct a non-commutative Fourier transform for probability measures on signatures of branched rough paths. We use the latter to provide sufficient conditions for a random signature to be determined by its expected value, thus giving an answer to the uniqueness moment problem for branched rough paths.