The geometry of controlled rough paths

Understand

We prove that the spaces of controlled (branched) rough paths of arbitrary order form a continuous field of Banach spaces.

• This structure has many similarities to an (infinite-dimensional) vector bundle and allows to define a topology on the total space, the collection of all controlled path spaces, which turns out to be Polish in the geometric case.
• The construction is intrinsic and based on a new approximation result for controlled rough paths.
• This framework turns well-known maps such as the rough integration map and the It\^o-Lyons map into continuous (structure preserving) mappings.