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arXiv · 2111.15539

Smooth rough paths, their geometry and algebraic renormalization

Carlo Bellingeri·Peter K. Friz·Sylvie Paycha·Rosa Preiß
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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.

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Carlo Bellingeri, Peter K. Friz, Sylvie Paycha, Rosa Preiß. 2021-11-30. Smooth rough paths, their geometry and algebraic renormalization. https://doi.org/10.1007/s10013-022-00570-7

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