Integration of branched rough paths

TL;DR

This paper develops a method to integrate branched rough paths with Lipschitz continuous one-forms, establishing bounds and continuity properties, and connecting it to associated $ ext{Pi}$-rough paths.

Contribution

It introduces a new integral construction for branched rough paths with Lipschitz one-forms, providing bounds and continuity results, and linking to $ ext{Pi}$-rough paths.

Findings

Constructed the integral of branched $p$-rough paths with Lipschitz one-forms.

Derived a quantitative bound for the integral.

Proved the integral depends continuously on the driving rough path.

Abstract

When the one-form is $Lip\left(\gamma-1\right) $ with $\gamma >p\geq 1$, we construct the integral of a branched $p$-rough path, which defines another branched $p$-rough path. We derive a quantitative bound for this integral and prove that it depends continuously on the driving branched rough path in rough path metric. Moreover, we prove that the first level branched rough integral coincides with a first level integral of the associated $\Pi$-rough path.