On the Signature of a Path in an Operator Algebra
We introduce a class of operators associated with the signature of a smooth path with values in a ⋆ algebra . These operators serve as the basis of Taylor expansions of solutions to controlled differential equations of interest in noncommutative probability. They are defined by fully contracting it...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2022 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2022
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211808 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the Signature of a Path in an Operator Algebra. Nicolas Gilliers and Carlo Bellingeri. SIGMA 18 (2022), 096, 43 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862574110379343872 |
|---|---|
| author | Gilliers, Nicolas Bellingeri, Carlo |
| author_facet | Gilliers, Nicolas Bellingeri, Carlo |
| citation_txt | On the Signature of a Path in an Operator Algebra. Nicolas Gilliers and Carlo Bellingeri. SIGMA 18 (2022), 096, 43 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We introduce a class of operators associated with the signature of a smooth path with values in a ⋆ algebra . These operators serve as the basis of Taylor expansions of solutions to controlled differential equations of interest in noncommutative probability. They are defined by fully contracting iterated integrals of , seen as tensors, with the product of . If it is considered that partial contractions should be included, we explain how these operators yield a trajectory on a group of representations of a combinatorial Hopf monoid. To clarify the role of partial contractions, we build an alternative group-valued trajectory whose increments embody full-contraction operators alone. We obtain, therefore, a notion of signature, which seems more appropriate for noncommutative probability.
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| first_indexed | 2026-03-13T13:41:00Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211808 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T13:41:00Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gilliers, Nicolas Bellingeri, Carlo 2026-01-12T10:13:52Z 2022 On the Signature of a Path in an Operator Algebra. Nicolas Gilliers and Carlo Bellingeri. SIGMA 18 (2022), 096, 43 pages 1815-0659 2020 Mathematics Subject Classification: 18M60; 18M80; 60L10; 46L89 arXiv:2102.11816 https://nasplib.isofts.kiev.ua/handle/123456789/211808 https://doi.org/10.3842/SIGMA.2022.096 We introduce a class of operators associated with the signature of a smooth path with values in a ⋆ algebra . These operators serve as the basis of Taylor expansions of solutions to controlled differential equations of interest in noncommutative probability. They are defined by fully contracting iterated integrals of , seen as tensors, with the product of . If it is considered that partial contractions should be included, we explain how these operators yield a trajectory on a group of representations of a combinatorial Hopf monoid. To clarify the role of partial contractions, we build an alternative group-valued trajectory whose increments embody full-contraction operators alone. We obtain, therefore, a notion of signature, which seems more appropriate for noncommutative probability. The authors thank Kurusch Ebrahimi-Fard for many enlightening discussions. CB is supported by the DFG Research Unit FOR2402, and NG is funded by a DAAD kurzstipendium. We thank the anonymous referees for their detailed reports. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Signature of a Path in an Operator Algebra Article published earlier |
| spellingShingle | On the Signature of a Path in an Operator Algebra Gilliers, Nicolas Bellingeri, Carlo |
| title | On the Signature of a Path in an Operator Algebra |
| title_full | On the Signature of a Path in an Operator Algebra |
| title_fullStr | On the Signature of a Path in an Operator Algebra |
| title_full_unstemmed | On the Signature of a Path in an Operator Algebra |
| title_short | On the Signature of a Path in an Operator Algebra |
| title_sort | on the signature of a path in an operator algebra |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211808 |
| work_keys_str_mv | AT gilliersnicolas onthesignatureofapathinanoperatoralgebra AT bellingericarlo onthesignatureofapathinanoperatoralgebra |