Regularity of the Ito-Lyons map

TL;DR

This paper proves the Frechet differentiability of the Ito-Lyons solution map in rough differential equations, enabling Taylor expansions and natural dynamics on path spaces, including Driver’s flow.

Contribution

It establishes the regularity of the solution map as a Banach space map, simplifying the analysis of rough differential equations and their associated dynamics.

Findings

Frechet differentiability of the Ito-Lyons map established

Enables Taylor-like expansions for solutions

Facilitates construction of natural dynamics on path spaces

Abstract

We show in this note that the Ito-Lyons solution map associated to a rough differential equation is Frechet differentiable when understood as a map between some Banach spaces of controlled paths. This regularity result provides an elementary approach to Taylor-like expansions of Inahama-Kawabi type for solutions of rough differential equations depending on a small parameter, and makes the construction of some natural dynamics on the path space over any compact Riemannian manifold straightforward, giving back Driver's flow as a particular case.