Categorial Independence and Lévy Processes
We generalize Franz's independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusio...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2022 |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2022
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211713 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Categorial Independence and Lévy Processes. Malte Gerhold, Stephanie Lachs and Michael Schürmann. SIGMA 18 (2022), 075, 27 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862729106441895936 |
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| author | Gerhold, Malte Lachs, Stephanie Schürmann, Michael |
| author_facet | Gerhold, Malte Lachs, Stephanie Schürmann, Michael |
| citation_txt | Categorial Independence and Lévy Processes. Malte Gerhold, Stephanie Lachs and Michael Schürmann. SIGMA 18 (2022), 075, 27 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We generalize Franz's independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. The obtained independence for morphisms is called categorial independence. We define categorial Lévy processes on every tensor category with an initial unit object and present a construction generalizing the reconstruction of a Lévy process from its convolution semigroup via the Daniell-Kolmogorov theorem. Finally, we discuss examples showing that many known independences from algebra as well as from (noncommutative) probability are special cases of categorial independence.
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| first_indexed | 2026-04-17T14:41:47Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211713 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T14:41:47Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gerhold, Malte Lachs, Stephanie Schürmann, Michael 2026-01-09T12:40:00Z 2022 Categorial Independence and Lévy Processes. Malte Gerhold, Stephanie Lachs and Michael Schürmann. SIGMA 18 (2022), 075, 27 pages 1815-0659 2020 Mathematics Subject Classification: 18D10; 60G20; 81R50 arXiv:1612.05139 https://nasplib.isofts.kiev.ua/handle/123456789/211713 https://doi.org/10.3842/SIGMA.2022.075 We generalize Franz's independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. The obtained independence for morphisms is called categorial independence. We define categorial Lévy processes on every tensor category with an initial unit object and present a construction generalizing the reconstruction of a Lévy process from its convolution semigroup via the Daniell-Kolmogorov theorem. Finally, we discuss examples showing that many known independences from algebra as well as from (noncommutative) probability are special cases of categorial independence. The authors are grateful to Uwe Franz, Michael Skeide, Tobias Fritz, Simeon Reich, Orr Shalit, and the anonymous referees for helpful comments on earlier drafts of this article. The work of MG and MS was supported by the German Research Foundation (DFG), project number 397960675. MG’s work was carried out partially during the tenure of an ERCIM ‘Alain Bensoussan’ Fellowship Programme. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Categorial Independence and Lévy Processes Article published earlier |
| spellingShingle | Categorial Independence and Lévy Processes Gerhold, Malte Lachs, Stephanie Schürmann, Michael |
| title | Categorial Independence and Lévy Processes |
| title_full | Categorial Independence and Lévy Processes |
| title_fullStr | Categorial Independence and Lévy Processes |
| title_full_unstemmed | Categorial Independence and Lévy Processes |
| title_short | Categorial Independence and Lévy Processes |
| title_sort | categorial independence and lévy processes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211713 |
| work_keys_str_mv | AT gerholdmalte categorialindependenceandlevyprocesses AT lachsstephanie categorialindependenceandlevyprocesses AT schurmannmichael categorialindependenceandlevyprocesses |