Algebraic renormalisation of regularity structures

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 coint-eraction. 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.
Type de document :
Pré-publication, Document de travail
A paraitre dans Inventiones Mathematicae. 2018
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Contributeur : Lorenzo Zambotti <>
Soumis le : mercredi 21 novembre 2018 - 11:53:40
Dernière modification le : mardi 19 mars 2019 - 01:19:05


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  • HAL Id : hal-01389938, version 2


M Bruned, M Hairer, L. Zambotti. Algebraic renormalisation of regularity structures. A paraitre dans Inventiones Mathematicae. 2018. 〈hal-01389938v2〉



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