Lipschitz geometry and rough paths - Boutaib, Y (2016)
Motivated by building a Lipschitz structure on the reachability set of a set of rough differential equations, we study first Lipschitz maps in full detail and in many aspects: we show embedding properties of Lipschitz spaces, study algebraic properties, local versus global behaviour of Lipschitz maps, give a version of the inverse function and the constant rank theorems and analyse flows of Lipschitz vector fields. Along the way, we give quantitative estimates, which is, after all, one of the main strengths of the Lipschitz geometry. Second, we combine our observations on the local properties of rough paths and Lipschitz maps to give a rather flexible structure on manifolds that allow the definition of rough paths on them and explain how the same process can be used to define a notion of coloured paths on manifolds, assuming that one can build a suitable functorial rule. Finally, we make use of those remarks to derive a quantitative condition that endow orbits of vector fields with the structure of a Lipschitz manifold and show, that under this condition, the space of solutions of rough differential equations driven by rough paths is the same as the one we obtain by driving the RDEs with smooth paths.