Translations of rough paths in combinatorial Hopf algebras

TL;DR

This paper extends the concept of translation in rough paths to any combinatorial Hopf algebra, providing a unified algebraic framework and simplifying conditions for translations, with applications to branched rough paths.

Contribution

It introduces a generalized notion of translation in rough paths over combinatorial Hopf algebras, linking it to cointeraction and simplifying the algebraic conditions.

Findings

Translation in rough paths is equivalent to cointeraction of bialgebras.

Reformulating translations via substitutions simplifies the algebraic conditions.

Concrete example provided for planarly branched rough paths.

Abstract

We generalize Bruned et.al.'s notion of translation in geometric and branched rough paths to a notion of translation in rough paths over any combinatorial Hopf algebra. We show that this notion of translation is equivalent to two bialgebras being in cointeraction, subject to certain additional conditions. We argue that reformulating translations in terms of substitutions, provides simpler conditions for the cointeraction formulation. For the special case where the translation can be obtained from a product, we show how to obtain a description of the dual coaction. As a concrete example, we describe translations in planarly branched rough paths.