2023 Fiscal Year Final Research Report
Construction of infinite invariant measures for dissipative random dynamical systems and application to infinite mixing
| Project/Area Number |
21K20330
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Multi-year Fund |
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| Review Section |
0201:Algebra, geometry, analysis, applied mathematics,and related fields
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| Research Institution | Kitami Institute of Technology |
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Principal Investigator |
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| Project Period (FY) |
2021-08-30 – 2024-03-31
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| Keywords | 絶対連続不変測度 / 物理的測度 / 混合性 / マルコフ作用素 / マルコフ作用素コサイクル / エルゴード性 |
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| Outline of Final Research Achievements |
Although one of the main purposes, namely obtaining general results for the existence of sigma-finite infinite invariant measures for dissipative systems, has not been accomplished yet, we have succeeded in characterising the finitude of ergodic absolutely continuous invariant probability measures, which are in particular physical measures, with the maximal support for (random) dynamical systems. From this result, we found that even if one considers random dynamical systems consisting of dissipative maps, it is not rare for them to admit conservative, ergodic and sigma-finite absolutely continuous invariant measures. Additionally, if we replace `ergodicity' with `mixing' in the above finitude property, we have a candidate which could characterises it, and we proceed with this project.
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| Free Research Field |
ランダム力学系理論
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| Academic Significance and Societal Importance of the Research Achievements |
力学系理論は，物体の運動をはじめとして生物の世代ごとの個体数や気候などの時間発展を記述する数理モデルなど，あらゆる分野で出現し，それらの性質を数学的に研究することは基本的な問題である．さらに力学系に揺らぎが加わったランダム力学系を考えることも現実の物理現象などへの応用上重要である．本研究では，力学系・ランダム力学系について，統計の基礎である大数の法則が成り立つという意味で良い確率が高々有限個存在するための，等価な条件を導入し，様々な具体例についても統計的性質とともに考察することができた．
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