Developments on Quantization and Quantum Information Analysis in terms of Infinite Dimensional Stochastic Analysis
| Project/Area Number |
17540136
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Meijo University |
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Principal Investigator |
SAITO Kimiaki Meijo University, Department of Mathematics, Professor, 理工学部, 教授 (90195983)
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| Co-Investigator(Kenkyū-buntansha) |
HIDA Takeyuki Meijo University, Department of Mathematics, Professor, 理工学部, 教授 (90022508)
SI Si Aichi Prefectural University, Faculty of Information Science, Associate Professor, 情報科学部, 助教授 (70269687)
HIBINO Yuji Saga University, Department of Mathematics, Associate Professor, 理工学部, 助教授 (50253589)
NISHI Kenjiro Meijo University, Department of Mathematics, Assistant Professor, 理工学部, 講師 (30076616)
HARA-MIMACHI Yuko Meijo University, Department of Mathematics, Assistant Professor, 理工学部, 講師 (00218629)
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| Project Period (FY) |
2005 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Keywords | Infinite dimensional stochastic processes / Quantization / Quantum Information Theory / Infinite dimensional Laplacians / Finance / Fractional Brownian motion / Quantum Information Analysis / Functional differential equation / 確率解析 / 確率過程量子化法 / ファインマン経路積分 / レヴィ過程 |
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| Research Abstract |
We deeply appreciate the grant for scientific research (term : academic years 2005, 2006) from JSPS. In this research, we considered a quantization and a new approach to a quantum information analysis by researching the infinite dimensional stochastic analysis jointly from major fields : probability theory, analysis, non-commutative geometry, number theory and computer science.
Main results which we obtained are the following 1) We constructed an infinite dimensional stochastic process associated with the Levy Laplacian on a space based on a stochastic process given by difference between two independent Levy processes. Moreover we gave a necessary and sufficient condition for eigenfunctions of the Levy Laplacian, which has some relation to the quantum decomposition. 2) We can take a nuclear space based on the Levy trace as a domain of the Levy Laplacian and prove that the Levy Laplacian is an infinitesimal generator of an infinite dimensional Wiener process on this space. The state spac
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e of this construction of the stochastic process is new one which is different from the Cesaro Hilbert space introduced by Professor Accardi. 3) Introducing the quantum Levy Laplacian we applied the construction of the infinite dimensional stochastic process generated by the Levy Laplacian in 1) to that of the quantum stochastic process. This implies that the quantum stochastic process can be studied as an infinite dimensional stochastic analysis. The construction of infinite dimensional stochastic processes which the author researched is connected with the quantum information analysis. This result is also connected with the quantum computation. 4) We obtained a relationship between the Levy Laplacian and an infinite dimensional Fractional Ornstein-Uhlenbeck process. This relationship is important to be applied the stochastic analysis based on the Levy Laplacian for the mathematical finance. Moreover we can extend this result to get a relationship between the Laplacian and a general infinite dimensional Ornstein-Uhlenbeck process. We also can obtain some relationship between the quantum Fractional Ornstein-Uhlenbeck process and the quantum Levy Laplacian.
By the above research, in particular, we have started a new joint work with Professor Accardi of Volterra Center in University of Rome, and a joint program between Department of Mathematics in our University and the Volterra Center. We also have started a new joint research on the Levy Laplacian on the abstract Wiener space with Professor Kuo of Louisiana State University in USA, which is developed to fruitful results in quantum theory. Less
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Report
(3 results)
Research Products
(28 results)
