Deformations of probability distributions on non-commutative probability spaces

2013 Fiscal Year Final Research Report

• Project/Area Number: 21540213
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Ochanomizu University
• Principal Investigator: YOSHIDA Hiroaki お茶の水女子大学, 大学院人間文化創成科学研究科, 教授 (10220667)
• Project Period (FY): 2009-04-01 – 2014-03-31
• Keywords: 関数解析学 / 作用素環論 / 非可換確率論 / 変形分布論 / 量子情報理論

Research Abstract

In this project, a deformed family of the classical Meixner distributions was considered as one of important examples for quantum deformations, and non-commutative random variables having these deformed Meixner distributions were constructed on the deformed Fock space by using the corresponding deformed annihilation and creation operators.
We have also investigated the fundamental theories on quantum deformed entropy and Fisher information. Especially, in the case of free probability, we have given another proofs for several inequalities on the relative free entropy and the free Fisher information based on the free diffusion processes.

Research Products

• [Journal Article] Random Matrices by MA models and compound free Poisson laws (2013)
  • Author(s): Ayako Hasegawa, Noriyoshi Sakuma and Hiroaki Yoshida
  • Journal Title: Probab. Math. Statist Volume: 33 Pages: 243-254
  • URL: http://www.math.uni.wroc.pl/~pms/files/33.2/Article/33.2.6.pdf
  • Peer Reviewed
• [Journal Article] New limit theorems related to free multiplicative convolution (2013)
  • Author(s): Noriyoshi Sakuma and Hiroaki Yoshida
  • Journal Title: Studia Math Volume: 214 Pages: 251-264
  • DOI: 10.4064/sm214-3-4
  • Peer Reviewed
• [Journal Article] An integral representation of the relative entropy (2012)
  • Author(s): Miku Hirata, Aya Nemoto, and Hiroaki Yoshida
  • Journal Title: Entropy Volume: 14 Pages: 1469–1477
  • DOI: 10.3390/e14081469
  • Peer Reviewed
• [Journal Article] The q-Meixner distributions associated with a q-deformed symmetric Fock space (2011)
  • Author(s): Hiroaki Yoshida
  • Journal Title: J. Phys. A : Math. Theor Volume: 44, 165306 Pages: 8
  • DOI: 10.1088/1751-8113/44/16/165306
  • Peer Reviewed
• [Journal Article] Remarks on the free relative entropy and the free Fisher information distance (2010)
  • Author(s): Hiroaki Yoshida
  • Journal Title: Infin.Dim. Anal. Quantum. Probab. Relat. Top Volume: 13 Pages: 243–255
  • DOI: 10.1142/S0219025710004048
  • Peer Reviewed
• [Presentation] The free logarithmic Sobolev and transportation cost inequalities by time integration (2013)
  • Author(s): Hiroaki Yoshida
  • Organizer: The 8th Jikji Workshop on Infinite Dimensional Analysis and Quantum Probability
  • Place of Presentation: National Iinstitute for Mathematical Sciences, Daejeon, Korea
  • Year and Date: 2013-08-20
  • Invited
• [Presentation] Applications of dissipation formulas of the relative free entropy (2013)
  • Author(s): Hiroaki Yoshida
  • Organizer: International Workshop on Noncommutative Analysis and its Future Prospects
  • Place of Presentation: Hokkaido University, Sapporo, Japan
  • Year and Date: 2013-08-06
  • Invited
• [Presentation] Applications of the time derivative of the Wasserstein distance and the free entropy dissipation (2013)
  • Author(s): Hiroaki Yoshida
  • Organizer: Workshop on Analytic, Stochastic, and Operator Algebraic Aspects of Noncommutative Distributions and Free Probability
  • Place of Presentation: Fields Institute, Univ. Toronto, Toronto, Canada
  • Year and Date: 2013-07-23
  • Invited
• [Presentation] On derivation of the relative free entropy and the Talagrand's inequality for the semi-circle law" (2012)
  • Author(s): Hiroaki Yoshida
  • Organizer: The 15th Workshop: Non-Commutative Harmonic Analysis
  • Place of Presentation: The Mathematical Research and Conference Center Banach Institute, Bedlewo, Poland
  • Year and Date: 2012-09-26
  • Invited
• [Presentation] Integral representa- tions of the relative free entropy associated with semicircular gradi- nts (2011)
  • Author(s): Hiroaki Yoshida
  • Organizer: The 14th Workshop Non-Commutative Harmonic Analysis
  • Place of Presentation: The Mathematical Research and Conference Center, Banach Institute, Bedlewo, Poland
  • Year and Date: 2011-09-27
  • Invited
• [Presentation] The free Poisson limit by the free Fisher information distance (2011)
  • Author(s): Hiroaki Yoshida
  • Organizer: Sapporo Workshop on Non-Commutative Analysis and Applica -tions to Complex Phenomena
  • Place of Presentation: Hokkaido Uniiversity, Sapporo, Japan
  • Year and Date: 2011-09-01
  • Invited
• [Presentation] On the free Fisher information distance and the logarithmic Sobolev inequality (2010)
  • Author(s): Hiroaki Yoshida
  • Organizer: The 13th Workshop NonCommutative Harmonic Analysis
  • Place of Presentation: The Mathematical Research and Conference Center Banach Institute,Bedlewo, Poland
  • Year and Date: 2010-07-14
  • Invited
• [Presentation] On self-adjoint q-Meixner operators (2010)
  • Author(s): Hiroaki Yoshida
  • Organizer: The 6th Jikji Workshop on Infinite Dimensional Analysis and Quantum Probability
  • Place of Presentation: National Iinstitute for Mathematical Sciences, Daejeon, Korea
  • Year and Date: 2010-01-10
  • Invited
• [Presentation] Remarks on the free analogue of relative entropy and of mutualinformation (2009)
  • Author(s): Hiroaki Yoshida
  • Organizer: Neumann-Zeminarium
  • Place of Presentation: Alfred Renyi Institue of Mathematics, Budapest, Hungary
  • Year and Date: 2009-11-10
  • Invited
• [Presentation] Linked partitions and deformed Meixner laws (2009)
  • Author(s): Hiroaki Yoshida
  • Organizer: The 12th Workshop : Non-Commutative Harmonic Analysis
  • Place of Presentation: The Mathematical Research and Conference Center Banach Institute, Bedlewo, Poland
  • Year and Date: 2009-08-20
  • Invited