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.
Highlights
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
Where the sum is indexed by the set of non-planar rooted trees τ ∈ TA with vertex decorations in the set A = {0, 1, . . . , d}
The Ff [τ ] are the so-called elementary differentials corresponding to the vector fields fi and the rooted tree τ ∈ TA with decorations in A
Summary
It is convenient to have a purely algebraic description of the flow, for this purpose we define the formal flow map φ ∈ HBΩ∗CK ⊗ ̄ HBΩCK φ :=. Given a pre-Lie algebra morphism g, we can attempt to find its adjoint g∗, defined such that. Suppose that the vector field f is replaced by an expansion in terms of pre-Lie products of f , for instance f h2 2 f. Collecting the powers in h appearing in (Ff⊗ Xst)(φ) results in a new series, which may be related algebraically to the. The above is generalised by first noting that it may be seen as the computation of an adjoint. The previous lemma amounts to the computation of the adjoint of g. Bruned et al [5] considered the problem of finding an adjoint in a more general rough path setting, establishing the following result. Let a be an infinitesimal character on HBACK and define g to be the pre-Lie algebra morphism g( l) =.
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