Banach fixed point and flow approach for rough analysis
Probability
2026-02-20 v1 Analysis of PDEs
Rings and Algebras
Abstract
In this paper, we show that the main algebraic assumption required to perform a fixed point argument for rough differential equations implies the algebraic assumption for the Bailleul flow approach. This assumption requires that the rough path associated with the equation is given by a Hopf algebra whose coproduct admits a cocycle and has a tree-like basis. We show that the Hopf algebra of multi-indices does not satisfy the cocycle condition. This is a rigorous result on the impossibility, observed in practice, of performing a fixed point argument for multi-indices rough paths and multi-indices in Regularity Structures.
Cite
@article{arxiv.2602.17437,
title = {Banach fixed point and flow approach for rough analysis},
author = {Yvain Bruned and Yingtong Hou and Paul Laubie and Zhicheng Zhu},
journal= {arXiv preprint arXiv:2602.17437},
year = {2026}
}