Constructing general rough differential equations through flow approximations

TL;DR

This paper develops a unified framework for constructing rough differential equations using flow approximations, extending the theory with algebraic structures and introducing aromatic rough paths.

Contribution

It introduces a novel algebraic approach to rough differential equations, including high-order expansions and a new class of aromatic rough paths.

Findings

Unified framework for rough differential equations via flow approximations

Extension of rough path theory to aromatic rough paths

High-order expansion results for various driving signals

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.