Hopf and pre-Lie algebras in regularity structures

TL;DR

This paper presents an algebraic framework using Hopf and pre-Lie algebras to systematically understand and compute renormalisation in singular SPDEs within the theory of regularity structures.

Contribution

It introduces a novel algebraic approach employing Hopf and pre-Lie algebras to describe renormalisation processes in regularity structures for SPDEs.

Findings

Describes positive and negative renormalisation via interacting Hopf algebras

Shows how to compute renormalised nonlinearities in SPDEs

Focuses on a simplified case with one equation and one noise

Abstract

These lecture notes aim to present the algebraic theory of regularity structures as developed in arXiv:1303.5113, arXiv:1610.08468, and arXiv:1711.10239. The main aim of this theory is to build a systematic approach to renormalisation of singular SPDEs; together with complementary analytic results, these works give a general solution theory for a wide class of semilinear parabolic singular SPDEs in the subcritical regime. We demonstrate how "positive" and "negative" renormalisation can be described using interacting Hopf algebras, and how the renormalised non-linearities in SPDEs can be computed. For the latter, of crucial importance is a pre-Lie structure on non-linearities on which the negative renormalisation group acts through pre-Lie morphisms. To show the main aspects of these results without introducing many notations and assumptions, we focus on a special case of the general theory in which there is only one equation and one noise. These lectures notes are an expansion of the material presented at a minicourse with the same title at the Master Class and Workshop "Higher Structures Emerging from Renormalisation" at the ESI, Vienna, in November 2021.