Notes on area operator, geometric 2-rough paths and Young integral when p^-1+q^-1=1

TL;DR

This paper investigates the properties of the area operator in rough path theory, extends Young integral to a critical case, and clarifies conditions for path enhancement into geometric 2-rough paths.

Contribution

It establishes the closability of the area operator under 2-rough norm, identifies the Riemann-Stieltjes integral as the unique enhancement candidate, and extends Young integral to a critical exponent case.

Findings

Area operator is closable on continuous paths with bounded variation.

Riemann-Stieltjes integral uniquely enhances paths with vanishing 2-variation.

Young integral extension applies at the critical exponent p^-1+q^-1=1.

Abstract

1.When equipped with 2-rough norm and restricted to continuous paths with bounded variation, the area operator is a closable unbounded operator. 2.The area defined through Riemann-Stieltjes integral is the only possible candidate to enhance a path with vanishing 2-variation into a geometric 2-rough path. 3.Young integral is extended to p^-1+q^-1=1 by assigning a finer scale continuity.