Renormalised singular stochastic PDEs

TL;DR

This paper introduces a new algebraic approach to renormalized singular stochastic PDEs, simplifying previous methods by avoiding extended decorations and encompassing a broad class of renormalization maps.

Contribution

It provides a novel proof of the renormalized system that bypasses extended decorations, highlighting algebraic properties related to preparation maps.

Findings

New proof of renormalized system avoiding extended decorations

Applicable to a large class of renormalization maps including BPHZ

Reveals algebraic properties linked to preparation maps

Abstract

Extended decorations on naturally decorated trees were introduced in the work of Bruned, Hairer and Zambotti on algebraic renormalization of regularity structures to provide a convenient framework for the renormalization of systems of singular stochastic PDEs within that setting. This non-dynamical feature of the trees complicated the analysis of the dynamical counterpart of the renormalization process. We provide a new proof of the renormalized system by-passing the use of extended decorations and working for a large class of renormalization maps, with the BPHZ renormalization as a special case. The proof reveals important algebraic properties connected to preparation maps.