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Integration of branched rough paths

Xinru Liu, Danyu Yang

TL;DR

This work extends rough path theory to branched $p$-rough paths by constructing the integral of a branched $p$-rough path against a Lip$(\gamma-1)$ one-form for $\gamma>p$, yielding a new branched $p$-rough path with a quantitative $p$-variation bound and continuous dependence on the driving path in the rough path metric. It introduces slowly-varying one-forms along branched rough paths and shows the integral can be built via the stability of effects under multiplication and iterated integration, culminating in a robust integration theory parallel to the geometric setting. A key contribution is proving that the first level of this branched rough integral coincides with the first-level integral of an associated $\Pi$-rough path, linking branched and $\Pi$-rough-path formalisms and enabling transfer of results between frameworks. The results enhance differentiability and stability analyses of branched rough-path driven systems and broaden applications to stochastic PDEs and related models where Ito-calculus-like structures arise.

Abstract

When the one-form is $Lip\left(γ-1\right) $ with $γ>p\geq 1$, 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 $Π$-rough path.

Integration of branched rough paths

TL;DR

This work extends rough path theory to branched -rough paths by constructing the integral of a branched -rough path against a Lip one-form for , yielding a new branched -rough path with a quantitative -variation bound and continuous dependence on the driving path in the rough path metric. It introduces slowly-varying one-forms along branched rough paths and shows the integral can be built via the stability of effects under multiplication and iterated integration, culminating in a robust integration theory parallel to the geometric setting. A key contribution is proving that the first level of this branched rough integral coincides with the first-level integral of an associated -rough path, linking branched and -rough-path formalisms and enabling transfer of results between frameworks. The results enhance differentiability and stability analyses of branched rough-path driven systems and broaden applications to stochastic PDEs and related models where Ito-calculus-like structures arise.

Abstract

When the one-form is with , we construct the integral of a branched -rough path, which defines another branched -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 -rough path.
Paper Structure (4 sections, 7 theorems, 53 equations)

This paper contains 4 sections, 7 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Notations and Definitions
  3. Main Result
  4. Proofs

Key Result

Lemma 12

For $p\geq 1$, suppose $X:\left[ 0,T\right] \rightarrow G_{d}^{\left[ p\right] }$ is a branched $p$-rough path. Let $\beta$ be a slowly-varying one-form along $X$ taking values in $\mathbb{R} ^{e}$ such that $\left\Vert \beta \right\Vert _{\theta }<\infty$ for some $\theta >1$. Define $y_{t}=\int_{0

Theorems & Definitions (15)

  • Definition 2: $p$-variation
  • Definition 3: branched $p$-rough path
  • Definition 7: $\Pi$-rough path
  • Definition 8
  • Definition 10: slowly-varying one-form
  • Definition 11: effect
  • Lemma 12
  • Lemma 15
  • Definition 16
  • Theorem 17
  • ...and 5 more