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 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.

Final version to appear in Inventiones Mathematicae

References in corpus (6)

• KPZ reloaded
• Renormalising SPDEs in regularity structures
• An analytic BPHZ theorem for regularity structures
• A simple construction of the continuum parabolic Anderson model on ${\mathbf{R}}^2$
• The motion of a random string
• The geometry of characters of Hopf algebras

Cited by in corpus (12)

• An analytic BPHZ theorem for regularity structures
• The reconstruction theorem in Besov spaces
• A solution theory for quasilinear singular SPDEs
• The motion of a random string
• Hopf-algebraic deformations of products and Wick polynomials
• Paracontrolled calculus and regularity structures
• The geometry of characters of Hopf algebras
• Martingale-driven approximations of singular stochastic PDEs
• Large-scale limit of interface fluctuation models
• Commutator estimates from a viewpoint of regularity structures
• An introduction to singular SPDEs
• Stochastic Heat Equations with Values in a Riemannian Manifold