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Algebraic renormalisation of regularity structures

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Bruned, Yvain, Hairer, Martin and Zambotti, L. (2019) Algebraic renormalisation of regularity structures. Inventiones Mathematicae, 215 (3). pp. 1039-1156. doi:10.1007/s00222-018-0841-x ISSN 0020-9910.

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Official URL: http://dx.doi.org/10.1007/s00222-018-0841-x

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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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Hopf algebras, Statistical mechanics, Stochastic analysis, Automorphisms
Journal or Publication Title: Inventiones Mathematicae
Publisher: Springer
ISSN: 0020-9910
Official Date: March 2019
Dates:
DateEvent
March 2019Published
13 December 2018Available
17 November 2018Accepted
Volume: 215
Number: 3
Page Range: pp. 1039-1156
DOI: 10.1007/s00222-018-0841-x
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Open Access (Creative Commons)
Date of first compliant deposit: 30 October 2019
Date of first compliant Open Access: 30 October 2019
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
UNSPECIFIEDLeverhulme Trusthttp://dx.doi.org/10.13039/501100000275
615897European Research Councilhttp://dx.doi.org/10.13039/501100000781
UNSPECIFIEDInstitut Universitaire de Francehttp://dx.doi.org/10.13039/501100004795
ANR-15-CE40-0020-01Agence Nationale de la Recherchehttp://dx.doi.org/10.13039/501100001665

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