Bogoliubov type recursions for renormalisation in regularity structures

TL;DR

This paper introduces Bogoliubov-type recursions within the framework of regularity structures to better understand renormalisation in singular SPDEs, connecting algebraic methods with stochastic analysis.

Contribution

It develops a new algebraic recursion approach for renormalisation in regularity structures, inspired by Connes-Kreimer's BPHZ formulation, and applies it to SPDEs.

Findings

Reformulation of renormalisation using Bogoliubov recursions

Application to specific SPDE renormalisation problems

Enhanced algebraic understanding of positive and negative renormalisation

Abstract

Hairer's regularity structures transformed the solution theory of singular stochastic partial differential equations. The notions of positive and negative renormalisation are central and the intricate interplay between these two renormalisation procedures is captured through the combination of cointeracting bialgebras and an algebraic Birkhoff-type decomposition of bialgebra morphisms. This work revisits the latter by defining Bogoliubov-type recursions similar to Connes and Kreimer's formulation of BPHZ renormalisation. We then apply our approach to the renormalisation problem for SPDEs.