The geometry of controlled rough paths
Abstract
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 (infinitedimensional) 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 wellknown maps such as the rough integration map and the ItôLyons map into continuous (structure preserving) mappings. Moreover, it is compatible with previous constructions of interest in the stability theory for rough integration.
 Publication:

arXiv eprints
 Pub Date:
 March 2022
 DOI:
 10.48550/arXiv.2203.05946
 arXiv:
 arXiv:2203.05946
 Bibcode:
 2022arXiv220305946G
 Keywords:

 Mathematics  Probability;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Rings and Algebras;
 34K50 (Primary);
 37H10;
 37H15;
 60H99;
 60G15 (Secondary)
 EPrint:
 28 pages