Characterizing measures according to their Radon–Nikodym cocycles: canonical marked Gibbs measures under the action of the diffeomorphism group
UDC 517.98 Suppose that we have a canonical Gibbs measure $\mu$ defined on a marked configuration space $\Omega$ that describes a system of infinitely many indistinguishable particles with internal degrees of freedom together with a diffeomorphism group action on $\Omega.$ Then $\mu$ is quasiinvaria...
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Institute of Mathematics, NAS of Ukraine
2026
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513239454973952 |
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| author | Kuna, Tobias Goldin, Gerald A. Kondratiev, Yuri G. Silva, José L. Kuna, Tobias Goldin, Gerald A. Kondratiev, Yuri G. Silva, José L. |
| author_facet | Kuna, Tobias Goldin, Gerald A. Kondratiev, Yuri G. Silva, José L. Kuna, Tobias Goldin, Gerald A. Kondratiev, Yuri G. Silva, José L. |
| author_sort | Kuna, Tobias |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-03-21T13:32:15Z |
| description | UDC 517.98
Suppose that we have a canonical Gibbs measure $\mu$ defined on a marked configuration space $\Omega$ that describes a system of infinitely many indistinguishable particles with internal degrees of freedom together with a diffeomorphism group action on $\Omega.$ Then $\mu$ is quasiinvariant under the group action, and we obtain a class of associated cocycles from its Radon–Nikodym derivatives. The cocycles are defined up to $\mu$-measure zero sets. We show that it is possible to choose a suitable pointwise-defined version $\beta$ of this cocycle. Further, we characterize all the measures on $\Omega$ that possess $\beta$ as their cocycle. If $\mu$ is obtained (e.g.) from a particular two-body potential $\hat{V}$ (satisfying some mild regularity assumptions), then $\beta$ takes a certain explicit form, and the class of canonical Gibbs measures characterized by $\beta$ contains exactly the measures associated with the potential $\hat{V}.$ Our result is based on the inheritance properties for the characterization by cocycles of Radon–Nikodym derivatives, which are proved for general $G$-spaces for local infinite-dimensional groups. |
| doi_str_mv | 10.3842/umzh.v77i4.8675 |
| first_indexed | 2026-03-24T03:41:31Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-8675 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:41:31Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-86752026-03-21T13:32:15Z Characterizing measures according to their Radon–Nikodym cocycles: canonical marked Gibbs measures under the action of the diffeomorphism group Characterizing measures according to their Radon–Nikodym cocycles: canonical marked Gibbs measures under the action of the diffeomorphism group Kuna, Tobias Goldin, Gerald A. Kondratiev, Yuri G. Silva, José L. Kuna, Tobias Goldin, Gerald A. Kondratiev, Yuri G. Silva, José L. Canonical Gibbs measures, marked Poisson measures, diffeomorphism groups, quasi-invariant measure, Radon-Nikodym derivatives UDC 517.98 Suppose that we have a canonical Gibbs measure $\mu$ defined on a marked configuration space $\Omega$ that describes a system of infinitely many indistinguishable particles with internal degrees of freedom together with a diffeomorphism group action on $\Omega.$ Then $\mu$ is quasiinvariant under the group action, and we obtain a class of associated cocycles from its Radon–Nikodym derivatives. The cocycles are defined up to $\mu$-measure zero sets. We show that it is possible to choose a suitable pointwise-defined version $\beta$ of this cocycle. Further, we characterize all the measures on $\Omega$ that possess $\beta$ as their cocycle. If $\mu$ is obtained (e.g.) from a particular two-body potential $\hat{V}$ (satisfying some mild regularity assumptions), then $\beta$ takes a certain explicit form, and the class of canonical Gibbs measures characterized by $\beta$ contains exactly the measures associated with the potential $\hat{V}.$ Our result is based on the inheritance properties for the characterization by cocycles of Radon–Nikodym derivatives, which are proved for general $G$-spaces for local infinite-dimensional groups. УДК 517.98 Характеризація мір за їхніми коциклами Радона–Нікодима: канонічні позначені гіббсівські міри під дією групи дифеоморфізмів Припустимо, що задано канонічну гіббсівську міру $\mu$ на просторі позначених конфігурацій $\Omega,$ яка описує систему нескінченної кількості нерозрізнюваних частинок із внутрішніми ступенями свободи, а також дію групи дифеоморфізмів на $\Omega.$ Тоді міра $\mu$ є квазіінваріантною щодо дії цієї групи, і з її похідних Радона–Нікодима утворюється клас відповідних коциклів. Коцикли визначено з точністю до $\mu$-нульових множин. Показано, що існує підходящий варіант цього коциклу $\beta,$ визначений поточково. Далі охарактеризовано всі міри на $\Omega,$ які мають $\beta$ як свій коцикл. Якщо міру $\mu$ отримано, наприклад, із певного двочастинкового потенціалу $\hat{V}$ (який задовольняє помірні умови регулярності), то $\beta$ набуває явної форми, а клас канонічних гіббсівських мір, охарактеризованих $\beta,$ точно збігається з мірами, пов'язаними з потенціалом $\hat{V}.$ Отримані результати базуються на властивостях успадкування для характеризації коциклами похідних Радона–Нікодима, доведених для загальних $G$-просторів із локальними нескінченновимірними групами. Institute of Mathematics, NAS of Ukraine 2026-03-21 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/8675 10.3842/umzh.v77i4.8675 Ukrains’kyi Matematychnyi Zhurnal; Vol. 77 No. 4 (2025); 286–287 Український математичний журнал; Том 77 № 4 (2025); 286–287 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/8675/10445 Copyright (c) 2025 Tobias Kuna, Gerald A. Goldin, Yuri G. Kondratiev, José L. Silva |
| spellingShingle | Kuna, Tobias Goldin, Gerald A. Kondratiev, Yuri G. Silva, José L. Kuna, Tobias Goldin, Gerald A. Kondratiev, Yuri G. Silva, José L. Characterizing measures according to their Radon–Nikodym cocycles: canonical marked Gibbs measures under the action of the diffeomorphism group |
| title | Characterizing measures according to their Radon–Nikodym cocycles: canonical marked Gibbs measures under the action of the diffeomorphism group |
| title_alt | Characterizing measures according to their Radon–Nikodym cocycles: canonical marked Gibbs measures under the action of the diffeomorphism group |
| title_full | Characterizing measures according to their Radon–Nikodym cocycles: canonical marked Gibbs measures under the action of the diffeomorphism group |
| title_fullStr | Characterizing measures according to their Radon–Nikodym cocycles: canonical marked Gibbs measures under the action of the diffeomorphism group |
| title_full_unstemmed | Characterizing measures according to their Radon–Nikodym cocycles: canonical marked Gibbs measures under the action of the diffeomorphism group |
| title_short | Characterizing measures according to their Radon–Nikodym cocycles: canonical marked Gibbs measures under the action of the diffeomorphism group |
| title_sort | characterizing measures according to their radon–nikodym cocycles: canonical marked gibbs measures under the action of the diffeomorphism group |
| topic_facet | Canonical Gibbs measures marked Poisson measures diffeomorphism groups quasi-invariant measure Radon-Nikodym derivatives |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8675 |
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