Smooth rough paths, their geometry and algebraic renormalization

TL;DR

This paper introduces smooth rough paths, emphasizing algebraic and geometric properties, and presents a Maurer-Cartan approach to renormalization and extension theorems in rough path theory.

Contribution

It develops a purely algebraic framework for rough path renormalization using Maurer-Cartan theory, extending to geometric, quasi-geometric, and Hopf algebraic settings.

Findings

Algebraic renormalization of rough paths via Maurer-Cartan perspective

Extension of rough path theory to quasi-geometric and Hopf algebraic contexts

Simplification of proofs by avoiding sewing arguments

Abstract

We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons extension theorem, the renormalization of rough paths in the spirit of [Bruned, Chevyrev, Friz, Prei{\ss}, A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019] as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.