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 series. For this purpose, we present a socalled arborification of the HoffmanIhara theory of quasishuffle 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 quasishuffle 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.
Original language  English 

Pages (fromto)  4363 
Number of pages  20 
Journal  Bulletin of the london mathematical society 
Volume  52 
Issue number  1 
DOIs  
Publication status  Published  6 Nov 2019 
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Yvain Bruned
 School of Mathematics  Lecturer in Mathematical Sciences
Person: Academic: Research Active (Teaching)