The Sewing lemma for $0 < \gamma \leq 1$

TL;DR

This paper extends the Sewing lemma to the regime 0 < γ ≤ 1, providing a continuous Sewing map that is not unique, with applications to rough path theory and Hopf algebras.

Contribution

It introduces a non-unique, continuous Sewing map for 0 < γ ≤ 1 and applies it to simplify the Lyons-Victoir extension theorem and analyze actions on rough paths.

Findings

Constructed a continuous Sewing map for 0 < γ ≤ 1.

Provided a simple constructive proof of the Lyons-Victoir extension theorem.

Established bicontinuity of a free action on the set of Rough Paths.

Abstract

We establish a Sewing lemma in the regime $\gamma \in \left( 0, 1 \right]$, constructing a Sewing map which is neither unique nor canonical, but which is nonetheless continuous with respect to the standard norms. Two immediate corollaries follow, which hold on any commutative graded connected locally finite Hopf algebra: a simple constructive proof of the Lyons-Victoir extension theorem which associates to a H\"older path a rough path, with the additional result that this map can be made continuous; the bicontinuity of a transitive free action of a space of H\"older functions on the set of Rough Paths.