Algebraic renormalisation of regularity structures

M Bruned (1) , M Hairer (1) , L Zambotti (2)

1 WMI - Warwick Mathematics Institute
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Domaines: Probabilités [math.PR]; Anneaux et algèbres [math.RA]; Equations aux dérivées partielles [math.AP]

M Bruned, M Hairer, L Zambotti. Algebraic renormalisation of regularity structures. 2016. ⟨hal-01389938v1⟩