Smooth rough paths, their geometry and algebraic renormalization
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| author | Bellingeri, Carlo Friz, Peter K. Paycha, Sylvie Preiß, Rosa |
| author_facet | Bellingeri, Carlo Friz, Peter K. Paycha, Sylvie Preiß, Rosa |
| contents | We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons extension theorem, the renormalization of rough paths in the spirit of [Bruned, Chevyrev, Friz, Preiß, A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019] as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_15539 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Smooth rough paths, their geometry and algebraic renormalization Bellingeri, Carlo Friz, Peter K. Paycha, Sylvie Preiß, Rosa Probability Classical Analysis and ODEs Rings and Algebras 60L20 (Primary) 60L70, 16T05, 22E66 (Secondary) We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons extension theorem, the renormalization of rough paths in the spirit of [Bruned, Chevyrev, Friz, Preiß, A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019] as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting. |
| title | Smooth rough paths, their geometry and algebraic renormalization |
| topic | Probability Classical Analysis and ODEs Rings and Algebras 60L20 (Primary) 60L70, 16T05, 22E66 (Secondary) |
| url | https://arxiv.org/abs/2111.15539 |