Renormalization: A Quasi-shuffle Approach

Menous, Frédéric; Patras, Frédéric

doi:10.1007/978-3-030-01593-0_21

Cited by 4 publications

(4 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As a final remark we mention that in the Connes–Kreimer approach to renormalisation in perturbative quantum field theory was studied from the point of view of commutative and non‐commutative quasi‐shuffle bialgebras.…”

Section: Introductionmentioning

confidence: 99%

Quasi‐shuffle algebras and renormalisation of rough differential equations

Bruned

Curry

Ebrahimi‐Fard

2019

Bull. London Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Quasi‐shuffle algebras and renormalisation of rough differential equations

Bruned

Curry

Ebrahimi‐Fard

2019

Bull. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…We note that quasi-shuffle structures have emerged independently in renormalization theory, see e.g. [17,39,45,46] and the references therein.…”

Section: Rough Paths and Renormalizationmentioning

confidence: 96%

“…We briefly review quasi-shuffle algebras in the version of [36]. Their origin can be traced back to [10] and we also refer the reader to [24,46]. We shall use Hoffman's isomorphism in the context of stochastic integration ('Itô vs. Stratonovich') which was discussed in detail in [23].…”

Section: Quasi-shuffle Algebrasmentioning

confidence: 99%

Smooth Rough Paths, Their Geometry and Algebraic Renormalization

et al. 2022

View full text Add to dashboard Cite

We introduce the class of “smooth rough paths” and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer–Cartan perspective is the key to a purely algebraic form of Lyons’ extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of “sum of rough paths”. We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.

show abstract

“…We briefly review quasi-shuffle algebras in the version of [Hof00]. Their origin can be traced back to [Car72] and we also refer the reader to [FP20,MP18]. We shall use Hoffman's isomorphism in the context of stochastic integration ('Itô vs. Stratonovich') which was discussed in detail in [EMPW15].…”

Section: Quasi-shuffle Algebrasmentioning

confidence: 99%

Smooth rough paths, their geometry and algebraic renormalization

Bellingeri¹,

Friz²,

Paycha³

et al. 2021

Preprint

View full text Add to dashboard Cite

We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons extension theorem, the renormalization of rough paths in the spirit of [Bruned, Chevyrev, Friz, Preiß, A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019] as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.

show abstract

Renormalization: A Quasi-shuffle Approach

Cited by 4 publications

References 37 publications

Quasi‐shuffle algebras and renormalisation of rough differential equations

Quasi‐shuffle algebras and renormalisation of rough differential equations

Smooth Rough Paths, Their Geometry and Algebraic Renormalization

Smooth rough paths, their geometry and algebraic renormalization

Contact Info

Product

Resources

About