An isomorphism between branched and geometric rough paths

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 Itô-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.

21 pages. Minor corrections. Accepted version to appear in Ann. Inst. H. Poincaré Probab. Statist

References in corpus (2)

• Renormalising SPDEs in regularity structures
• Quasi-shuffle algebras and renormalisation of rough differential equations

Cited by in corpus (6)

• The geometry of the space of branched Rough Paths
• Quasi-shuffle algebras and renormalisation of rough differential equations
• Renormalisation from non-geometric to geometric rough paths
• Branched Itô formula and natural Itô-Stratonovich isomorphism
• A remainder estimate for branched rough differential equations
• Rough path theory