Algebraic renormalisation of regularity structures

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Title: Algebraic renormalisation of regularity structures
Authors: Bruned, Y
Hairer, M
Zambotti, L
Item Type: Journal Article
Abstract: We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms. This subgroup is sufficiently large to be able to implement a version of the BPHZ renormalisation prescription in this context. This is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms, which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations. Our construction is based on bialgebras of decorated coloured forests in cointeraction. More precisely, we have two Hopf algebras in cointeraction, coacting jointly on a vector space which represents the generalised functions of the theory. Two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic Birkhoff factorisation that arises naturally in perturbative quantum field theory.
Issue Date: 18-Dec-2018
Date of Acceptance: 4-Nov-2018
ISSN: 0020-9910
Publisher: Springer Verlag
Journal / Book Title: Inventiones Mathematicae
Copyright Statement: © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Sponsor/Funder: The Leverhulme Trust
Commission of the European Communities
Funder's Grant Number: RL-2012-020-Transfer In
Keywords: 0101 Pure Mathematics
General Mathematics
Publication Status: Published online
Open Access location:
Online Publication Date: 2018-12-13
Appears in Collections:Pure Mathematics

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