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 (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 Itô-Lyons map into continuous (structure preserving) mappings. Moreover, it is compatible with previous constructions of interest in the stability theory for rough integration.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2022
- DOI:
- 10.48550/arXiv.2203.05946
- arXiv:
- arXiv:2203.05946
- Bibcode:
- 2022arXiv220305946G
- Keywords:
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- Mathematics - Probability;
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Rings and Algebras;
- 34K50 (Primary);
- 37H10;
- 37H15;
- 60H99;
- 60G15 (Secondary)
- E-Print:
- 28 pages