Integration of geometric rough paths

TL;DR

This paper bridges rough path theory with noncommutative algebra, extending integration concepts to include time-varying integrands through a novel class of slowly-varying one-forms.

Contribution

It introduces a non-abelian Young integration framework by connecting rough paths with noncommutative algebra, enabling natural handling of time-varying integrands.

Findings

Established a non-abelian Young integration model.

Identified a stable class of slowly-varying one-forms.

Extended rough path theory to include time-varying integrands.

Abstract

We build a connection between rough path theory and noncommutative algebra, and interpret the integration of geometric rough paths as an example of a non-abelian Young integration. We identify a class of slowly-varying one-forms, and prove that the class is stable under basic operations. In particular rough path theory is extended to allow a natural class of time varying integrands.