{
  "id": "1801.02964",
  "version": "v1",
  "published": "2018-01-09T14:47:55.000Z",
  "updated": "2018-01-09T14:47:55.000Z",
  "title": "Quasi-shuffle algebras and renormalisation of rough differential equations",
  "authors": [
    "Yvain Bruned",
    "Charles Curry",
    "Kurusch Ebrahimi-Fard"
  ],
  "categories": [
    "math.PR",
    "math.CA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-09T14:47:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rough differential equations",
      "renormalisation",
      "substitution bialgebra",
      "hopf algebra structures",
      "quasi-shuffle algebra automorphisms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}