The non-linear sewing lemma constructs flows of rough differential equations from a broad class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential equations, in the spirit of the backward error analysis. Mixing algebra and analysis, a Taylor formula with remainder and a composition formula are central in the expansion analysis. With a suitable algebraic structure on the non-smooth vector fields to be integrated, we recover in a single framework several results regarding high-order expansions for various kinds of driving paths. We also extend the notion of driving rough path. We introduce as an example a new family of branched rough paths, called aromatic rough paths modeled after aromatic Butcher series.
The author thanks Laure Coutin and Antoine Brault for interesting discussions related to this article, and the referees for their valuable comments.
"Constructing general rough differential equations through flow approximations." Electron. J. Probab. 27 1 - 24, 2022. https://doi.org/10.1214/21-EJP717