Algebraic renormalisation of regularity structures

TL;DR

This paper introduces a systematic algebraic framework for renormalising stochastic PDEs with nonlinearities involving generalized functions, utilizing new regularity structures and Hopf algebra techniques to streamline the process.

Contribution

It develops a new class of regularity structures with a large automorphism group, enabling a direct algebraic BPHZ renormalisation approach for stochastic PDEs.

Findings

Constructed a new regularity structure with an explicit automorphism subgroup.

Implemented a BPHZ renormalisation scheme within this algebraic framework.

Connected the theory to Hopf algebras and algebraic Birkhoff factorisation.

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.