Banach fixed point and flow approach for rough analysis

TL;DR

This paper explores the algebraic structures necessary for fixed point methods in rough differential equations, demonstrating limitations with certain Hopf algebras and clarifying when these approaches are feasible.

Contribution

It establishes the algebraic conditions needed for fixed point arguments in rough analysis and shows the Hopf algebra of multi-indices does not meet these conditions.

Findings

Hopf algebra of multi-indices lacks cocycle property

Fixed point methods are limited for certain rough paths

Clarifies algebraic constraints in rough differential equations

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.