Regularity of the Ito-Lyons map

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.

Final version, 11 pages. An integration by parts formula for the Ito-Lyons map has been added as another application of the main result

References in corpus (2)

• Smoothness of the density for solutions to Gaussian rough differential equations
• Two-parameter stochastic calculus and Malliavin's integration-by-parts formula on Wiener space

Cited by in corpus (1)

• The inverse problem for rough controlled differential equations