Deformation of Bargmann-Fock representation in non-commutative probability theory

• Project/Area Number: 16K05175
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Aichi University of Education
• Principal Investigator: Asai Nobuhiro 愛知教育大学, 教育学部, 教授 (60399029)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: フォック空間 / 非可換確率論 / 直交多項式論 / 確率分布 / B型ガウス・ポアソン作用素 / 変形フォック空間 / 直交多項式 / 離散確率分布 / 変形ガウス型作用素 / ポアソン型作用素 / 変形マイクスナー分布 / 荷重フォック空間 / 量子確率論 / バーグマン測度 / 関数解析学 / 確率論

Outline of Final Research Achievements

We considered deformations of the Bargmann-Fock space, Gaussian/Poisson operators, and their probability distributions in terms of non-commutative probabilistic approach. (1) The radial Bargmann density function associated with the Fock space of type B is explicitly obtained. (2) Gaussian-Poisson type operators on the type B space are introduced. As a result, we showed that a q2-Meixner class of probability measures can be treated within the framework of type B. (3) We constructed a weighted q-Fock space and presented unknown and non-trivial examples in previous works.

Academic Significance and Societal Importance of the Research Achievements

本研究において，従来のフォック空間ならびにガウス・ポアソン型作用素の変形・補間理論の拡張により，確率論や非可換確率論では馴染みの薄い非自明な確率分布とより強い非可換性との関係解明への足掛かりになっていることが意義深い．

Research Products

• [Int'l Joint Research] オハイオ州立大学(米国)
• [Int'l Joint Research] オハイオ州立大学(米国)
• [Int'l Joint Research] オハイオ州立大学(米国)
• [Int'l Joint Research] University of Wroclaw(ポーランド)
• [Int'l Joint Research] Chungbuk National University/Dankook University(韓国)
• [Int'l Joint Research] ブロツワフ大学(ポーランド)
• [Journal Article] Deformed Gaussian Operators on Weighted q-Fock Spaces (2020)
  • Author(s): Nobuhiro Asai、Hiroaki Yoshida
  • Journal Title: Journal of Stochastic Analysis Volume: 1 (4) Issue: 4 Pages: 6-6
  • DOI: 10.31390/josa.1.4.06
  • Peer Reviewed / Open Access
• [Journal Article] On (α,q)-Poisson Operators and Their Distributions (2020)
  • Author(s): Nobuhiro Asai
  • Journal Title: RIMS Kokyuroku, RIMS, Kyoto University Volume: -
  • NAID: 120007166004
• [Journal Article] Poisson type operators on the Fock space of type B (2019)
  • Author(s): Nobuhiro Asai and Hiroaki Yoshida
  • Journal Title: Journal of Mathematical Physics Volume: 60 Issue: 1 Pages: 011702-011702
  • DOI: 10.1063/1.5074114
  • NAID: 120006732119
  • Peer Reviewed
• [Journal Article] On weighted Fock spaces and distributions (2018)
  • Author(s): Nobuhiro Asai
  • Journal Title: RIMS Kokyuroku, RIMS, Kyoto University Volume: 2089 Pages: 84-94
  • Open Access
• [Journal Article] On weighted Fock spaces and distributions. (2018)
  • Author(s): Nobuhiro Asai
  • Journal Title: RIMS Kokyuroku, RIMS, Kyoto University Volume: 印刷中
  • Open Access
• [Journal Article] Radial Density Function Associated with the (\alpha,q)-Fock Space (2017)
  • Author(s): Nobuhiro Asai, M. Bozejko and T. Hasebe
  • Journal Title: MI Lecture note Volume: 72
  • Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Interpolations of Bargmann type measures (2016)
  • Author(s): Nobuhiro Asai, A. D. Krystek and L. J. Wojakowski
  • Journal Title: Demonstr. Math. Volume: 49 Issue: 3
  • DOI: 10.1515/dema-2016-0025
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Radial Bargmann representation for the Fock space of type B (2016)
  • Author(s): Nobuhiro Asai, Marek Bozejko and Takahiro Hasebe
  • Journal Title: Journal of Mathematical Physics Volume: 57 Issue: 2 Pages: 021702-021702
  • DOI: 10.1063/1.4939748
  • NAID: 120006661339
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] On weighted q-Fock spaces (2021)
  • Author(s): Nobuhiro Asai
  • Organizer: 京都大学数理解析研究所共同研究（公開型）量子場の数理とその周辺
  • Invited
• [Presentation] The Boas-Buck's classical problem on generating functions (2020)
  • Author(s): Nobuhiro Asai
  • Organizer: Workshop on Non-commutative Probability and Related Fields
  • Invited
• [Presentation] Some Gaussian type operators and nontrivial examples from q-orthogonal polynomials (2019)
  • Author(s): Nobuhiro Asai
  • Organizer: Workshop on ``Non-commutative Probability and Related Fields"
  • Int'l Joint Research / Invited
• [Presentation] Deformed Fock spaces and Operators (2019)
  • Author(s): Nobuhiro Asai
  • Organizer: The QBIC Workshop 2019
  • Int'l Joint Research / Invited
• [Presentation] Gaussian-Poisson field operator on the Fock space of type B and q^2 -deformed Meixner distribution (2019)
  • Author(s): Nobuhiro Asai
  • Organizer: The 40th International Conference on Quantum Probability and Infinite Dimensional Analysis
  • Int'l Joint Research / Invited
• [Presentation] Deformed free Fock spaces and Meixner distributions (2019)
  • Author(s): Nobuhiro Asai
  • Organizer: 京都大学数理解析研究所共同研究 ( 公開型 ) 「量子場の数理とその周辺」
  • Int'l Joint Research / Invited
• [Presentation] Probability distributions of a field operator associated with a q-deformed algebra (2019)
  • Author(s): Nobuhiro Asai
  • Organizer: AMS-MAA Joint Mathematical Meeting, AMS Special session on Orthogonal Polynomials, Quantum Probability, Harmonic Analysis, and Stochastic Analysis
  • Int'l Joint Research / Invited
• [Presentation] Poisson type operator on the Fock space of type B (2018)
  • Author(s): Nobuhiro Asai
  • Organizer: Non-commutative probability and related fields
  • Int'l Joint Research / Invited
• [Presentation] Weighted Fock spaces with multiparameter and corresponding probability distributions (2018)
  • Author(s): Nobuhiro Asai
  • Organizer: AMS-MAA Joint Mathematical Meeting, Special session on Orthogonal Polynomials, Quantum Probability, and Stochastic Analysis
  • Int'l Joint Research / Invited
• [Presentation] Deformed generating functions and orthogonal polynomials (2017)
  • Author(s): Nobuhiro Asai
  • Organizer: Workshop on Quantum Probability, Infinite Dimensional Analysis and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Analytic characterization theorems in infinite dimensional distribution theory (2017)
  • Author(s): Nobuhiro Asai
  • Organizer: 京都大学数理解析研究所 共同研究(公開型)「量子場の数理とその周辺」
  • Invited
• [Presentation] Radial density function associated to the Fock space of type B and Rogers-Szego polynomials (2016)
  • Author(s): Nobuhiro Asai
  • Organizer: Workshop on Infinite Dimensional Analysis and Quantum Probability
  • Place of Presentation: 清州市（韓国）
  • Int'l Joint Research / Invited
• [Presentation] Levy processes and operator algebras, with applications (2016)
  • Author(s): Nobuhiro Asai
  • Organizer: 17th Workshop: Non-Commutative probability-- Levy processes and operator algebras, with applications
  • Place of Presentation: ベドレボ（ポーランド）
  • Int'l Joint Research / Invited
• [Presentation] The radial Bargmann measure for the Fock space of type B (2016)
  • Author(s): Nobuhiro Asai
  • Organizer: 九州大学マス・フォア・インダストリ研究所研究集会：量子場の数理とその周辺
  • Place of Presentation: 九州大学（福岡県福岡市）
  • Invited
• [Funded Workshop] Workshop on “Non-commutative Probability and Related Fields” (2019)
• [Funded Workshop] AMS Special session on Orthogonal Polynomials, Quantum Probability, Harmonic Analysis, and Stochastic Analysis (2019)