Algebraic renormalisation of regularity structures

Yvain Bruned, M. Hairer, L. Zambotti

Research output: Contribution to journalArticlepeer-review

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.
Original languageEnglish
Pages (from-to)1039-1156
Number of pages118
JournalInventiones mathematicae
Volume215
Issue number3
Early online date13 Dec 2018
DOIs
Publication statusPublished - Mar 2019

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