Gibbs measure over the cone of vector-valued discrete measures
UDC 517.9 We consider a gas each particle of which is characterized by a pair $(x,v_x),$ where $x\in \mathbb R^d$ is the position and $v_x\in \mathbb R^d_0=\mathbb R^d\setminus \{0\}$ is the velocity. Gibbs measures are defined on the cone of vector-valued measures. Our aim is to prove their existen...
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| Datum: | 2026 |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/8201 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512972069142528 |
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| author | Di Persio, Luca Kondratiev, Yuri Vardanyan , Viktorya Di Persio, Luca Kondratiev, Yuri Vardanyan , Viktorya |
| author_facet | Di Persio, Luca Kondratiev, Yuri Vardanyan , Viktorya Di Persio, Luca Kondratiev, Yuri Vardanyan , Viktorya |
| author_sort | Di Persio, Luca |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-03-21T13:31:23Z |
| description | UDC 517.9
We consider a gas each particle of which is characterized by a pair $(x,v_x),$ where $x\in \mathbb R^d$ is the position and $v_x\in \mathbb R^d_0=\mathbb R^d\setminus \{0\}$ is the velocity. Gibbs measures are defined on the cone of vector-valued measures. Our aim is to prove their existence. We introduce a family of probability measures $\mu_\lambda$ on the cone $\mathbb K(\mathbb R^d)$ and define local Hamiltonian and partition functions for a positive, symmetric, bounded, and measurable pair potential. By using the definitions mentioned above, we define Gibbs measure as a solution to the Dobrushin–,Lanford–Ruelle equation. In particular, we focus on the subset of tempered Gibbs measures. To prove the existence of the Gibbs measure, we show that the subset of tempered Gibbs measures is nonempty and relatively compact. |
| doi_str_mv | 10.3842/umzh.v77i4.8201 |
| first_indexed | 2026-03-24T03:37:16Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-8201 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:37:16Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-82012026-03-21T13:31:23Z Gibbs measure over the cone of vector-valued discrete measures Gibbs measure over the cone of vector-valued discrete measures Di Persio, Luca Kondratiev, Yuri Vardanyan , Viktorya Di Persio, Luca Kondratiev, Yuri Vardanyan , Viktorya Interacting particle systems Vector valued Radon measures DLR equation Gibbs measure Local Gibbs specification Local Hamiltonian and partition functions for a positive, symmetric, bounded and measurable pair potential Gibbs’s measure as a solution to the Dobrushin-Lanford-Ruelle equation. tempered Gibbs measures Pair potential, Hamiltonian UDC 517.9 We consider a gas each particle of which is characterized by a pair $(x,v_x),$ where $x\in \mathbb R^d$ is the position and $v_x\in \mathbb R^d_0=\mathbb R^d\setminus \{0\}$ is the velocity. Gibbs measures are defined on the cone of vector-valued measures. Our aim is to prove their existence. We introduce a family of probability measures $\mu_\lambda$ on the cone $\mathbb K(\mathbb R^d)$ and define local Hamiltonian and partition functions for a positive, symmetric, bounded, and measurable pair potential. By using the definitions mentioned above, we define Gibbs measure as a solution to the Dobrushin–,Lanford–Ruelle equation. In particular, we focus on the subset of tempered Gibbs measures. To prove the existence of the Gibbs measure, we show that the subset of tempered Gibbs measures is nonempty and relatively compact. УДК 517.9 Гіббсівська міра на конусі векторнозначних дискретних мір Розглянуто газ, у якому кожну частинку охарактеризовано парою $(x, v_x)$, де $x \in \mathbb{R}^d$ – положення, а $v_x \in \mathbb{R}^d_0 = \mathbb{R}^d \setminus {0}$ – ненульова швидкість. Гіббсівські міри визначено на конусі векторнозначних мір. Метою роботи є доведення їхнього існування. Запропоновано сім’ю ймовірнісних мір $\mu_\lambda$ на конусі $\mathbb{K}(\mathbb{R}^d)$, а також означено локальний гамільтоніан і статистичну суму для додатного, симетричного, обмеженого та вимірного парного потенціалу. На основі цих означень побудовано гіббсівську міру як розв'язок рівняння Добрушина–Ленфорда–Рюелля. Особливу увагу приділено підмножині темперованих гіббсівських мір. Для доведення існування гіббсівської міри встановлено, що ця підмножина є непорожньою та відносно компактною. Institute of Mathematics, NAS of Ukraine 2026-03-21 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/8201 10.3842/umzh.v77i4.8201 Ukrains’kyi Matematychnyi Zhurnal; Vol. 77 No. 4 (2025); 282–283 Український математичний журнал; Том 77 № 4 (2025); 282–283 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/8201/10443 Copyright (c) 2025 Luca Di Persio, Yuri Kondratiev, Viktorya Vardanyan |
| spellingShingle | Di Persio, Luca Kondratiev, Yuri Vardanyan , Viktorya Di Persio, Luca Kondratiev, Yuri Vardanyan , Viktorya Gibbs measure over the cone of vector-valued discrete measures |
| title | Gibbs measure over the cone of vector-valued discrete measures |
| title_alt | Gibbs measure over the cone of vector-valued discrete measures |
| title_full | Gibbs measure over the cone of vector-valued discrete measures |
| title_fullStr | Gibbs measure over the cone of vector-valued discrete measures |
| title_full_unstemmed | Gibbs measure over the cone of vector-valued discrete measures |
| title_short | Gibbs measure over the cone of vector-valued discrete measures |
| title_sort | gibbs measure over the cone of vector-valued discrete measures |
| topic_facet | Interacting particle systems Vector valued Radon measures DLR equation Gibbs measure Local Gibbs specification Local Hamiltonian and partition functions for a positive symmetric bounded and measurable pair potential Gibbs’s measure as a solution to the Dobrushin-Lanford-Ruelle equation. tempered Gibbs measures Pair potential Hamiltonian |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8201 |
| work_keys_str_mv | AT dipersioluca gibbsmeasureovertheconeofvectorvalueddiscretemeasures AT kondratievyuri gibbsmeasureovertheconeofvectorvalueddiscretemeasures AT vardanyanviktorya gibbsmeasureovertheconeofvectorvalueddiscretemeasures AT dipersioluca gibbsmeasureovertheconeofvectorvalueddiscretemeasures AT kondratievyuri gibbsmeasureovertheconeofvectorvalueddiscretemeasures AT vardanyanviktorya gibbsmeasureovertheconeofvectorvalueddiscretemeasures |