A Rough Path Perspective on Renormalization

TL;DR

This paper introduces an algebraic framework for rough path translation, focusing on branched rough paths and their relation to regularity structures, with implications for algebraic renormalization theory.

Contribution

It develops a novel algebraic approach to rough path translation, connecting branched rough paths with regularity structures and algebraic renormalization.

Findings

Establishes algebraic structures for rough path translation.

Links branched rough paths with Hairer's regularity structures.

Provides new insights into algebraic renormalization theory.

Abstract

We develop the algebraic theory of rough path translation. Particular attention is given to the case of branched rough paths, whose underlying algebraic structure (Connes-Kreimer, Grossman-Larson) makes it a useful model case of a regularity structure in the sense of Hairer. Pre-Lie structures are seen to play a fundamental rule which allow a direct understanding of the translated (i.e. renormalized) equation under consideration. This construction is also novel with regard to the algebraic renormalization theory for regularity structures due to Bruned--Hairer--Zambotti (2016), the links with which are discussed in detail.