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, Preiß, 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.

Rough paths and SPDE

“…We mention in particular the paper [11], where iterated Itô integrals of a semimartingale are seen as paths with values in the character group of a specific quasi-shuffle algebra, see [56]. We also recall that smooth paths with values in character groups of a specific class of Hopf algebras have several interesting properties, see [13].…”