Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

An isomorphism between branched and geometric rough paths

Horatio Boedihardjo and Ilya Chevyrev

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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 [J. Differential Equations 225 (2006) 103–133] as well as a canonical version of the Itô–Stratonovich correction formula of Hairer–Kelly [Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015) 207–251]. 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.


Nous explicitons un isomorphisme naturel entre les espaces de chemins rugueux branchants et géométriques. Ceci fournit une généralisation multi-échelle de l’isomorphisme de Lejay–Victoir [J. Differential Equations 225 (2006) 103–133], ainsi qu’une version canonique de la formule pour le terme correctif d’Itô–Stratonovich d’Hairer et Kelly [Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015) 207–251]. Notre construction est élémentaire et utilise la propriété que l’algèbre de Grossman–Larson est isomorphe à une algèbre tensorielle.

Nous appliquons cet isomorphisme pour étudier la signature des chemins rugueux branchants. Plus précisément, nous montrons que la signature d’un chemin rugueux branchant est triviale si et seulement si le chemin a une structure arborescente, et nous construisons une transformée de Fourier non commutative pour les mesures de probabilités sur les signatures de chemins rugueux branchants. Nous utilisons cette dernière pour donner des conditions suffisantes pour qu’une signature aléatoire soit déterminée par sa valeur moyenne, fournissant ainsi une réponse au problème d’unicité des moments pour les chemins rugueux branchants.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 1131-1148.

Received: 22 December 2017
Revised: 21 April 2018
Accepted: 24 April 2018
First available in Project Euclid: 14 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 16T05: Hopf algebras and their applications [See also 16S40, 57T05] 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

Branched rough paths Butcher group Signature Non-commutative Fourier transform


Boedihardjo, Horatio; Chevyrev, Ilya. An isomorphism between branched and geometric rough paths. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 1131--1148. doi:10.1214/18-AIHP912.

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