Rough differential equations and planarly branched universal limit theorem

TL;DR

This paper extends the universal limit theorem in rough path theory to planarly branched rough paths with roughness between 1/4 and 1/3, using Banach fixed point methods, generalizing previous results.

Contribution

It proves the universal limit theorem for planarly branched rough paths with specific roughness, broadening the scope of rough path theory.

Findings

Universal limit theorem established for planarly branched rough paths

Extension to roughness 1/4 < α ≤ 1/3

Application of Banach fixed point theorem

Abstract

The universal limit theorem is a central result in rough path theory, which has been proved for: (i) rough paths with roughness $\frac{1}{3}< \alpha \leq \frac{1}{2}$; (ii) geometric rough paths with roughness $0< \alpha \leq 1$; (iii) branched rough paths with roughness $0< \alpha \leq 1$. Planarly branched rough paths are natural generalizations of both rough paths and branched rough paths, in the sense that post-Lie algebras are generalizations of both Lie algebras and pre-Lie algebras. Here the primitive elements of the graded dual Hopf algebra of the Hopf algebra corresponding to the planarly branched rough paths (resp. rough paths, resp. branched rough paths) form a post-Lie (resp. Lie, resp. pre-Lie algebra). In this paper, we prove the universal limit theorem for planarly branched rough paths with roughness $\frac{1}{4}< \alpha \leq \frac{1}{3}$, via the method of Banach fixed point theorem.