Smooth rough paths, their geometry and algebraic renormalization
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.
Cite
@article{arxiv.2111.15539,
title = {Smooth rough paths, their geometry and algebraic renormalization},
author = {Carlo Bellingeri and Peter K. Friz and Sylvie Paycha and Rosa Preiß},
journal= {arXiv preprint arXiv:2111.15539},
year = {2024}
}
Comments
47 pages