Quasi-shuffle algebras and renormalisation of rough differential equations

TL;DR

This paper explores the use of quasi-shuffle algebras and Hopf algebra structures to compare different renormalisation techniques in rough differential equations, connecting algebraic methods with stochastic analysis.

Contribution

It introduces an arborification of Hoffman's quasi-shuffle automorphisms and links them to renormalisation procedures in rough path theory, unifying algebraic and stochastic approaches.

Findings

Canonical renormalisation aligns with Marcus' extension for semimartingales.

Arborification provides a new algebraic perspective on rough path renormalisation.

Recursive formulas relate different rough path constructions via Hopf algebra coactions.

Abstract

The objective of this work is to compare several approaches to the process of renormalisation in the context of rough differential equations using the substitution bialgebra on rooted trees known from backward error analysis of $B$-series. For this purpose, we present a so-called arborification of the Hoffman--Ihara theory of quasi-shuffle algebra automorphisms. The latter are induced by formal power series, which can be seen to be special cases of the cointeraction of two Hopf algebra structures on rooted forests. In particular, the arborification of Hoffman's exponential map, which defines a Hopf algebra isomorphism between the shuffle and quasi-shuffle Hopf algebra, leads to a canonical renormalisation that coincides with Marcus' canonical extension for semimartingale driving signals. This is contrasted with the canonical geometric rough path of Hairer and Kelly by means of a recursive formula defined in terms of the coaction of the substitution bialgebra.