Renormalisation from non-geometric to geometric rough paths

TL;DR

This paper compares a new renormalisation method for rough paths with existing ones, demonstrating their compatibility and providing explicit formulas, thereby advancing the understanding of geometric and non-geometric rough path transformations.

Contribution

It establishes the commutation of the new renormalisation with the Hairer-Kelly map and compares it with other renormalisation techniques in rough path theory.

Findings

Renormalisation commutes with the Hairer-Kelly map.

Explicit formulas for the renormalisation are derived.

Behavior of renormalisation in alternative approaches is analyzed.

Abstract

The Hairer-Kelly map has been introduced for establishing a correspondence between geometric and non-geometric rough paths. Recently, a new renormalisation on rough paths has been proposed in (arxiv 1810.12179), built on this map and the Lyons-Victoir extension theorem. In this work, we compare this renormalisation with the existing ones such as BPHZ and the local products renormalisations. We prove that they commute in a certain sense with the Hairer-Kelly map and exhibit an explicit formula in the framework of (arxiv 1810.12179). We also see how the renormalisation behaves in the alternative approach in ( arXiv:1712.01965) for moving from non-geometric to geometric rough paths.