Rough differential equations and planarly branched universal limit theorem

Nannan Li; Xing Gao; Dominique Manchon

doi:10.3934/jcd.2026013

Article Contents

2026, Volume 16: 1-45. Doi: 10.3934/jcd.2026013

This volume Previous Article Variational integrators for stochastic Hamiltonian systems on Lie groups: properties and convergence Next Article A brief introduction to matrix hydrodynamics

Rough differential equations and planarly branched universal limit theorem

1.
School of Mathematics and Statistics, Lanzhou University Lanzhou, 730000, China
2.
Laboratoire de Mathématiques Blaise Pascal, CNRS-Université Clermont-Auvergne, 3 place Vasarély, CS 60026, 63178 Aubière, France

^*Corresponding author: Dominique Manchon
^*Corresponding author: Dominique Manchon

Received: July 24, 2025

Revised: June 22, 2026

Published online: July 10, 2026

The third author thanks Lanzhou University for warm welcome and the excellent working atmosphere in which this article has been achieved.

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Abstract

We prove the universal limit theorem for planarly branched rough paths with roughness $ \frac{1}{4} < \alpha \leq \frac{1}{3} $, using a fixed-point approach based on the Banach contraction principle. Planarly branched rough paths generalize both classical rough paths and branched rough paths, in a manner analogous to how post-Lie algebras generalize both Lie and pre-Lie algebras. In particular, the primitive elements in the graded dual of the Hopf algebra associated with planarly branched rough paths form a post-Lie algebra, subsuming the Lie and pre-Lie structures arising in the geometric and branched settings. This result extends the scope of the universal limit theorem, previously established for: (ⅰ) rough paths with roughness $ \frac{1}{3} < \alpha \leq \frac{1}{2} $; (ⅱ) geometric rough paths with $ 0 < \alpha \leq 1 $; and (ⅲ) branched rough paths with $ 0 < \alpha \leq 1 $.

Keywords:

Planarly branched rough path,
controlled planarly branched rough path,
universal limit theorem,
Banach fixed point theorem,
planar rooted forest.

Mathematics Subject Classification: Primary: 60L20, 60L50; Secondary: 60H99, 34K50, 37H10, 05C05.

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