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arXiv · 2006.10309

Constructing general rough differential equations through flow approximations

Antoine Lejay
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Abstract

The non-linear sewing lemma constructs flows of rough differential equations from a braod 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 kind of driving paths. We also extend the notion of driving rough path. We also introduce as an example a new family of branched rough paths, called aromatic rough paths modeled after aromatic Butcher series.

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Antoine Lejay. 2020-06-18. Constructing general rough differential equations through flow approximations. https://arxiv.org/abs/2006.10309

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