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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.
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https://hal.archives-ouvertes.fr/hal-01389938
Contributor : Lorenzo Zambotti <>
Submitted on : Wednesday, November 21, 2018 - 11:53:40 AM
Last modification on : Friday, March 27, 2020 - 3:06:19 AM

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  • HAL Id : hal-01389938, version 2

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M Bruned, M Hairer, L. Zambotti. Algebraic renormalisation of regularity structures. 2018. ⟨hal-01389938v2⟩

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