Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases
We study the hydrodynamic limits of three kinds of one-dimensional stochastic log-gases known as Dyson's Brownian motion model, its chiral version, and the Bru-Wishart process studied in dynamical random matrix theory. We define the measure-valued processes so that their Cauchy transforms solve...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| 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/123456789/211619 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases. Taiki Endo, Makoto Katori and Noriyoshi Sakuma. SIGMA 18 (2022), 049, 22 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862728269619527680 |
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| author | Endo, Taiki Katori, Makoto Sakuma, Noriyoshi |
| author_facet | Endo, Taiki Katori, Makoto Sakuma, Noriyoshi |
| citation_txt | Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases. Taiki Endo, Makoto Katori and Noriyoshi Sakuma. SIGMA 18 (2022), 049, 22 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study the hydrodynamic limits of three kinds of one-dimensional stochastic log-gases known as Dyson's Brownian motion model, its chiral version, and the Bru-Wishart process studied in dynamical random matrix theory. We define the measure-valued processes so that their Cauchy transforms solve the complex Burgers-type equations. We show that applications of the method of characteristic curves to these partial differential equations provide the functional equations relating the Cauchy transforms of measures at an arbitrary time with those at the initial time. We transform the functional equations for the Cauchy transforms to those for the -transforms and the -transforms of the measures, which play central roles in free probability theory. The obtained functional equations for the -transforms and the -transforms are simpler than those for the Cauchy transforms and are useful for explicit calculations, including the computation of free cumulant sequences. Some of the results are argued using the notion of free convolutions.
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| first_indexed | 2026-04-17T14:28:29Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211619 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T14:28:29Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Endo, Taiki Katori, Makoto Sakuma, Noriyoshi 2026-01-07T13:39:04Z 2022 Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases. Taiki Endo, Makoto Katori and Noriyoshi Sakuma. SIGMA 18 (2022), 049, 22 pages 1815-0659 2020 Mathematics Subject Classification: 82C22; 60B20; 44A15; 46L54 arXiv:2106.00442 https://nasplib.isofts.kiev.ua/handle/123456789/211619 https://doi.org/10.3842/SIGMA.2022.049 We study the hydrodynamic limits of three kinds of one-dimensional stochastic log-gases known as Dyson's Brownian motion model, its chiral version, and the Bru-Wishart process studied in dynamical random matrix theory. We define the measure-valued processes so that their Cauchy transforms solve the complex Burgers-type equations. We show that applications of the method of characteristic curves to these partial differential equations provide the functional equations relating the Cauchy transforms of measures at an arbitrary time with those at the initial time. We transform the functional equations for the Cauchy transforms to those for the -transforms and the -transforms of the measures, which play central roles in free probability theory. The obtained functional equations for the -transforms and the -transforms are simpler than those for the Cauchy transforms and are useful for explicit calculations, including the computation of free cumulant sequences. Some of the results are argued using the notion of free convolutions. The present authors would like to thank Shinji Koshida and Yoshimichi Ueda for their useful comments on the manuscript. They are grateful to the anonymous referees for valuable suggestions for future studies on this subject. MK was supported by the Grant-in-Aid for Scientific Research (C) (No. 19K03674), (B) (No. 18H01124), (S) (No. 16H06338), (A) (No. 21H04432) of Japan Society for the Promotion of Science. NS was supported by the Grant-in-Aid for Scientific Research (B) (No. 19H01791) and (C)(No. 19K03515) of the Japan Society for the Promotion of Science. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases Article published earlier |
| spellingShingle | Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases Endo, Taiki Katori, Makoto Sakuma, Noriyoshi |
| title | Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases |
| title_full | Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases |
| title_fullStr | Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases |
| title_full_unstemmed | Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases |
| title_short | Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases |
| title_sort | functional equations solving initial-value problems of complex burgers-type equations for one-dimensional log-gases |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211619 |
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