Theoretical study of random dynamical systems through the approach of stochastic processes

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K21834
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: Osaka University (2023); Kyoto University (2019-2022)
• Principal Investigator: Yano Kouji 大阪大学, 大学院理学研究科, 教授 (80467646)
• Co-Investigator(Kenkyū-buntansha): 佐藤 譲 北海道大学, 電子科学研究所, 准教授 (30342794) ; 角 大輝 京都大学, 人間・環境学研究科, 教授 (40313324) ; 中野 雄史 東海大学, 理学部, 准教授 (50778313) ; 中村 文彦 北見工業大学, 工学部, 准教授 (40825147) ; 豊川 永喜 九州大学, マス・フォア・インダストリ研究所, 助教 (30907762)
• Project Period (FY): 2019-06-28 – 2024-03-31
• Keywords: ランダム力学系 / 確率過程論 / 一般化逆正弦法則 / 情報系分解 / 雑音誘起現象

Outline of Final Research Achievements

Among the various aspects of random dynamical systems, which are given by adding certain randomness to deterministic dynamical systems, we have paid particular attention to the similarities with various properties of Markov processes in the theory of stochastic processes, and have aimed to mathematically elucidate several particular properties of random dynamical systems that are produced by the complex interplay of the properties of the deterministic dynamical systems. In this research, we have obtained remarkable results on the arcsine and the Darling Kac laws for some random dynamical systems constructed by random choice of interval maps, the aging effects for skew Bessel processes, and the problem of resolution of the sigma fields for action evolutions into the driving noise, the infinite past noise, and the third noise.

Free Research Field

確率過程論

Academic Significance and Societal Importance of the Research Achievements

決定的力学系にランダム要素を加えたランダム力学系の性質を数学的に解明することは、学術的にも社会的にも重要な意義がある．学術的には，逆正弦法則，エイジング効果，情報系分解という重要なテーマについて，新しい研究発展の方向性を開拓することができた．社会的には，複雑かつ予測不能なランダム現象に潜む数理的構造を理論的に捉える新しい知見を得ることができた．