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...
Збережено в:
| Опубліковано в: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
| ISSN: | 1815-0659 |
| 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| Резюме: | 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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| ISSN: | 1815-0659 |