Translations of rough paths in combinatorial Hopf algebras

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. As a concrete example, we describe translations in planarly branched rough paths.

Smooth Rough Paths, Their Geometry and Algebraic Renormalization

“…More generally we hope to investigate in future work, what properties one should impose on D and v in a general Hopf algebra framework. A first suggestion of such a set of axioms was very recently given in [52,Definitions 6 and 7].…”

“…Remark 4.13 Concerning the question of defining translation maps for more general Hopf algebras H, the example of the free pre-Lie algebra points towards the general approach of using some 'free' algebraic structure on the Lie algebra P (H * ), if it is available, to define the translation map as a universal homomorphism of the free object. Ideas to this regard were proposed in [51, Definition 5.2.1] and [52,Definition 8]. In the latter, this basic idea is also applied to the case of the free post-Lie algebra taking the role of P (H * ).…”

Smooth rough paths, their geometry and algebraic renormalization

“…Concerning the question of defining translation maps for more general Hopf algebras H, the example of the free pre-Lie algebra points towards the general approach of using some 'free' algebraic structure on the Lie algebra P (H * ), if it is available, to define the translation map as a universal homomorphism of the free object. Ideas to this regard were proposed in [Pre21, Defn.5.2.1] and [Rah21,Defn.8]. In the latter, this basic idea is also applied to the case of the free post-Lie algebra taking the role of P (H * ).…”