2023 Fiscal Year Final Research Report
Correspondence between non-commutative probability, probability and univalent function theories
| Project/Area Number |
19K14546
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 12010:Basic analysis-related
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| Research Institution | Hokkaido University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | 単調独立性 / 分枝過程 / ランダム行列 / マルコフ過程 / 加法過程 / Loewner chain / 自由確率論 |
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| Outline of Final Research Achievements |
This research succeeded in enlarging the application of monotone independence to other fields. In addition to a known correspondence between Loewner chains and unitary monotone additive processes, we also constructed a correspondence to additive processes on the unit circle. Motivated by this work, we discovered a Loewner chain structure in branching processes that are known as a stochastic model for the population change. Consequently, we obtained a new method based on complex analysis to analyze the expectation and extinction probability of branching processes. Combining cyclic-monotone independence and free independence, we introduced a framework that allows us to treat random matrices with perturbation.
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| Free Research Field |
非可換確率論
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| Academic Significance and Societal Importance of the Research Achievements |
本研究は確率過程，非可換確率過程，複素関数論，ランダム行列といった分野を横断する研究である．本研究の成果としてこれらの分野どうしに新たな結びつきが生まれ，分野どうしが互いに交流を深めるという学術的な意義があった．特に，ランダム行列は幅広く科学に応用されている．自由確率論はランダム行列を解析する一つの大きな手法になっており，応用先として既に量子情報理論や深層学習理論がある．本研究では摂動を含むようなランダム行列の扱いに対する基本的な枠組みを提案しており，将来的にランダム行列のさらなる応用を目指していく上で参考になったり役に立ちうると考えている．
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