Parametrization of renormalized models for singular stochastic PDEs

TL;DR

This paper provides an explicit parametrization of admissible models for singular stochastic PDEs using the paracontrolled approach, and describes how renormalization schemes act on this parametrization.

Contribution

It introduces a linear parametrization of models in regularity structures and explicitly characterizes the action of renormalization schemes, including the BHZ scheme.

Findings

Explicit description of the action of renormalization schemes on model parametrization

Simplification of the action for degree preserving preparation maps

Application to the BHZ renormalization scheme

Abstract

Let $\mathscr{T}$ be the regularity structure associated with a given system of singular stochastic PDEs. The paracontrolled representation of the $\sf \Pi$ map provides a linear parametrization of the nonlinear space of admissible models $\sf M=(g,\Pi)$ on $\mathscr{T}$, in terms of the family of para-remainders used in the representation. We give an explicit description of the action of the most general class of renormalization schemes presently available on the parametrization space of the space of admissible models. The action is particularly simple for renormalization schemes associated with degree preserving preparation maps; the BHZ renormalization scheme has that property.