An isomorphism between branched and geometric rough paths

Author: 

Boedihardjo, H
Chevyrev, I

Publication Date: 

May 2019

Journal: 

ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES

Last Updated: 

2019-10-18T11:16:49.217+01:00

Issue: 

2

Volume: 

55

DOI: 

10.1214/18-AIHP912

page: 

1131-1148

abstract: 

© Association des Publications de l’Institut Henri Poincaré, 2019. 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.

Symplectic id: 

844238

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Submitted to ORA: 

Submitted

Publication Type: 

Journal Article