A top-down approach to algebraic renormalization in regularity structures based on multi-indices

TL;DR

This paper introduces a top-down algebraic framework for renormalization in regularity structures of semi-linear stochastic PDEs, utilizing multi-indices and a generalized Hopf algebra to construct canonical models.

Contribution

It develops a novel algebraic approach based on multi-indices and extends the Hopf algebra of derivations to facilitate renormalization in regularity structures.

Findings

Generalized Hopf algebra for derivations

Canonical smooth model construction

Renormalization via exponential map

Abstract

We provide an algebraic framework to describe renormalization in regularity structures based on multi-indices for a large class of semi-linear stochastic PDEs. This framework is ``top-down", in the sense that we postulate the form of the counterterm and use the renormalized equation to build a canonical smooth model for it. The core of the construction is a generalization of the Hopf algebra of derivations in [LOT23], which is extended beyond the structure group to describe the model equation via an exponential map: This allows to implement a renormalization procedure which resembles the preparation map approach in our context.