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arXiv · 2509.23148

Controlled rough paths: a general Hopf-algebraic setting

Zhicheng Zhu·Xing Gao·Nannan Li·Dominique Manchon
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Abstract

We set up controlled rough paths for a class of combinatorial Hopf algebras, encompassing shuffle, Butcher-Connes-Kreimer and Munthe-Kaas--Wright Hopf algebras. The class of controls we consider encompasses both H\"older continuous paths and (not necessarily continuous) paths with bounded $p$-variation. We prove existence and uniqueness of the solution of a lifted initial value problem in this general setting by applying the fixed point method in a suitable Banach space of controlled rough paths, and we prove a universal limit theorem addressing the robustness of the solution with respect to the parameters and the initial condition.

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Zhicheng Zhu, Xing Gao, Nannan Li, Dominique Manchon. 2025-09-27. Controlled rough paths: a general Hopf-algebraic setting. https://arxiv.org/abs/2509.23148

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