The geometry of controlled rough paths
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908469637939200 |
|---|---|
| author | Varzaneh, Mazyar Ghani Riedel, Sebastian Schmeding, Alexander Tapia, Nikolas |
| author_facet | Varzaneh, Mazyar Ghani Riedel, Sebastian Schmeding, Alexander Tapia, Nikolas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_05946 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The geometry of controlled rough paths Varzaneh, Mazyar Ghani Riedel, Sebastian Schmeding, Alexander Tapia, Nikolas Probability Classical Analysis and ODEs Rings and Algebras 34K50 (Primary), 37H10, 37H15, 60H99, 60G15 (Secondary) 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. |
| title | The geometry of controlled rough paths |
| topic | Probability Classical Analysis and ODEs Rings and Algebras 34K50 (Primary), 37H10, 37H15, 60H99, 60G15 (Secondary) |
| url | https://arxiv.org/abs/2203.05946 |