2003 Fiscal Year Annual Research Report

量子ホワイトノイズの非線形理論と非ガウス拡張

Research Project

Project/Area Number	15340039
Research Category	Grant-in-Aid for Scientific Research (B)
Research Institution	Tohoku University
Principal Investigator	尾畑伸明東北大学, 大学院・情報科学研究科, 教授 (10169360)
Co-Investigator(Kenkyū-buntansha)	洞彰人岡山大学, 環境理工学部, 助教授 (10212200) 浦川肇東北大学, 大学院・情報科学研究科, 教授 (50022679) 日合文雄東北大学, 大学院・情報科学研究科, 教授 (30092571) 齊藤公明名城大学, 理工学部, 教授 (90195983) 有光敏彦筑波大学, 物理学系, 教授 (50134200)
Keywords	ホワイトノイズ理論 / 量子確率論 / 量子ホワイトノイズ / 量子分解 / 量子中心極限定理 / レヴィ・ラプラシアン / 散逸量子系 / グラフのスペクトル
Research Abstract	伊藤解析に代表される伝統的な確率解析の枠組みではブラウン運動B(t)が基礎になるが、本研究の出発点はブラウン運動をさらに分解(量子分解+時間微分)して得られる「量子ホワイトノイズ」である。その超関数的な側面からは「非線形理論」を、代数的な側面からは「非ガウス拡張」をテーマとして浮かび上がらせ、その数理を探求することが研究目標である。そのために、個別的な共同研究に加えて、課題探索のためのワークショップの開催や国際会議などにおける研究発表を行った。具体的な項目と研究概略は以下の通り。(1)非線形ホワイトノイズ方程式:量子ホワイトノイズを基礎とする作用素論を発展させ、解の正則性を調べた。特に、新しい作用素微分を導入し、応用として、量子マルチンゲールの表現における積分核を導く公式を得た。(2)複素ホワイトノイズによる特徴づけ定理:シーガル・バーグマン写像を用いて、非核型空間におけるS変換と作用素シンボルの特徴づけ定理を証明した。(3)レヴィラプラシアンとホワイトノイズの高次べき:レヴィラプラシアンに付随する熱方程式から確率過程を構成し、その非可換拡張を研究している。(4)多モード相互作用フォック空間と非ガウス型確率過程の量子分解:グラフのスペクトル解析に関連する相互作用フォック空間と直交多項式を研究している。グラフの構造と独立性の諸概念の関係を見出し、量子中心極限定理のモデルを構成した。(5)無限次元調和解析への応用:量子ホワイトノイズによる作用素展開定理を精密化し、無限次元群のユニタリー表現の既約性の証明への応用を検討している。(6)量子系の物理への応用:散逸系を記述する量子確率微分方程式の構造を非平衡統計力学の手法を用いて解析した。

Research Products

(32 results)

All Other

All Publications (32 results)

[Publications] Y.Hashimoto, A.Hora, N.Obata: "Central limit theorems for large graphs : Method of quantum decomposition"J.Math.Phys.. 44. 71-88 (2003)
[Publications] A.Hora, N.Obata: "Quantum decomposition and quantum central limit theorem"Fundamental Problems in Quantum Mechanics(L.Accardi and S.Tasaki, Eds.),"World Scientific. 284-305 (2003)
[Publications] Un Cig Ji, N.Obata: "A unified characterization theorem in white noise theory"Infin.Dimen.Anal.Quantum Probab.Rel.Top.. 6. 167-178 (2003)
[Publications] N.Obata, H.Ouerdiane: "Heat equation associated with Levy Laplacian"Proc.Internat.Conf.Stochastic Analysis and Applications(S. Albeverio, A.B.de Monvel and H.Ouerdiane(Eds.)Kluwer. 53-68 (2004)
[Publications] Un Cig Ji, N.Obata: "A role of Bargmann--Segal spaces in characterization and expansion of operators on Fock space"J.Math.Soc.Japan. 56(印刷中). (2004)
[Publications] Un Cig Ji, N.Obata: "Unitarity of Kuo's Fourier--Mehler transform"Infin.Dimen.Anal.Quantum Probab.Rel.Top.. 7(印刷中). (2004)
[Publications] N.Obata: "Quantum probabilistic approach to spectral analysis of star graphs"Interdisciplinary Information Sciences. 10(印刷中). (2004)
[Publications] F.Hiai, Y.Ueda: "Automorphisms of free product-type and their crossed-products"J.Operator Theory. 50. 119-130 (2003)
[Publications] F.Hiai: "q-deformed Araki-Woods algebras"Operator Algebras and Mathematical Physics(J.-M.Combes et al.eds.),Bucharest. 169-202 (2003)
[Publications] F.Hiai, X.Zhan: "Submultiplicativity vs subadditivity for unitarily invariant norms"Linear Algebra Appl.. 377. 155-164 (2004)
[Publications] F.Hiai, D.Petz, Y.Ueda: "Free transportation cost inequalities via random matrix approximation"probability Theory and Related Fields. (印刷中).
[Publications] H.Urakawa: "The Cheeger constant, the heat kernel and the Green kernel of an infinite graph"Monatshefte fur Math.. 138. 225-237 (2003)
[Publications] S.Dragomir, T.Ichiyama, H.Urakawa: "Yang-Mills theory and conjugate connections"Differ.Geom.Its Appl.. 18. 229-238 (2003)
[Publications] N.Matsuura, H.Urakawa: "Discrete improper affine spheres"J.Geometry Physics. 45. 164-183 (2003)
[Publications] H.Urakawa: "Yang-Mills theory over compact symplectic manifolds"Annals Global Analysis Geometry. (印刷中).
[Publications] H.Urakawa: "The heat kernel and the Green kernel of an infinite graph"Contemporary Math.Amer.Math.Soc.. (印刷中). (2004)
[Publications] H.Urakawa: "The Dirichet eigenvalue problem, the finite element method and graph theory"Contemporary Math.Amer.Math.Soc.. (印刷中). (2004)
[Publications] A.Hora: "Scaling limit for Gibbs states of Johnson graphs and resulting Meixner classes"Infin.Dimen.Anal.Quantum Probab.Rel.Top.. 6. 139-143 (2003)
[Publications] A.Hora: "A noncommutative version of Kerov's Gaussian limit for the Plancherel measure of the symmetric group"Lecture Notes in Math.. 1815. 77-88 (2003)
[Publications] T.Arimitsu: "Quantum stochastic differential equations in view of non-equilibrium thermo field dynamics"Fundamental Aspects of Quantum Physics(L.Accardi and S.Tasaki, Eds.). 24-47 (2003)
[Publications] T.Arimitsu, N.Arimitsu: "Analysis of accelerations in turbulence based on generalized statistics"Condensed Matter Phys.. 6. 85-92 (2003)
[Publications] P.Jizba, T.Arimitsu: "The world according to Renyi : Thermodynamics of fractal systems"Annals of Physics. (印刷中).
[Publications] T.Arimitsu, N.Arimitsu: "Multifractal analysis of fluid particle accelerations in turbulence"Physica D. (印刷中). (2003)
[Publications] A.E.Kobryn, T.Hayashi, T.Arimitsu: "Quantum stochastic differential equations for Boson and Fermion systems ---Method of non-equilibrium thermo field dynamics"Annals of Physics. 308. 395-446 (2003)
[Publications] A.E.Kobryn, T.Hayashi, T.Arimitsu: "Quantum stochastic processed : Boson and Fermion Brownian motion"Condensed Matter Phys.. 6. 367-646 (2003)
[Publications] P.Jizba, T.Arimitsu: "On observability of Renyi's entropy"Phys.Rev.E. (印刷中).
[Publications] A.E.Kobryn, T.Hayashi, T.Arimitsu: "Boson and Fermion Brownian motion"J.Phys.Soc.Japan. 72. 58-61 (2003)
[Publications] N.Obata, K.Saito: "Cauchy processes and the Levy Laplacian"Quantum Probability and White Noise Analysis. 14. 360-373 (2003)
[Publications] K.Nishi, K.Saito, A.H.Tsoi: "Fractional Browian motions and the Levy Laplacian"QUantum Information. 5(印刷中). (2004)
[Publications] K.Nishi, K.Saito: "An infinite dimensional stochastic process and the Levy Laplacian acting on WND-valued functions"Quantum Information and Complexity. (印刷中). (2004)
[Publications] 明出伊類似, 尾畑伸明: "量子確率論の基礎"牧野書店. 296 (2003)
[Publications] F.Hiai, H.Kosaki: "Means of Hilbert Space Operators"Springer-Verlag. 148 (2003)

2003 Fiscal Year Annual Research Report

量子ホワイトノイズの非線形理論と非ガウス拡張

Principal Investigator

尾畑 伸明 東北大学, 大学院・情報科学研究科, 教授 (10169360)

Research Products

[Publications] Y.Hashimoto, A.Hora, N.Obata: "Central limit theorems for large graphs : Method of quantum decomposition"J.Math.Phys.. 44. 71-88 (2003)

[Publications] A.Hora, N.Obata: "Quantum decomposition and quantum central limit theorem"Fundamental Problems in Quantum Mechanics(L.Accardi and S.Tasaki, Eds.),"World Scientific. 284-305 (2003)

[Publications] Un Cig Ji, N.Obata: "A unified characterization theorem in white noise theory"Infin.Dimen.Anal.Quantum Probab.Rel.Top.. 6. 167-178 (2003)

[Publications] N.Obata, H.Ouerdiane: "Heat equation associated with Levy Laplacian"Proc.Internat.Conf.Stochastic Analysis and Applications(S. Albeverio, A.B.de Monvel and H.Ouerdiane(Eds.)Kluwer. 53-68 (2004)

[Publications] Un Cig Ji, N.Obata: "A role of Bargmann--Segal spaces in characterization and expansion of operators on Fock space"J.Math.Soc.Japan. 56(印刷中). (2004)

[Publications] Un Cig Ji, N.Obata: "Unitarity of Kuo's Fourier--Mehler transform"Infin.Dimen.Anal.Quantum Probab.Rel.Top.. 7(印刷中). (2004)

[Publications] N.Obata: "Quantum probabilistic approach to spectral analysis of star graphs"Interdisciplinary Information Sciences. 10(印刷中). (2004)

[Publications] F.Hiai, Y.Ueda: "Automorphisms of free product-type and their crossed-products"J.Operator Theory. 50. 119-130 (2003)

[Publications] F.Hiai: "q-deformed Araki-Woods algebras"Operator Algebras and Mathematical Physics(J.-M.Combes et al.eds.),Bucharest. 169-202 (2003)

[Publications] F.Hiai, X.Zhan: "Submultiplicativity vs subadditivity for unitarily invariant norms"Linear Algebra Appl.. 377. 155-164 (2004)

[Publications] F.Hiai, D.Petz, Y.Ueda: "Free transportation cost inequalities via random matrix approximation"probability Theory and Related Fields. (印刷中).

[Publications] H.Urakawa: "The Cheeger constant, the heat kernel and the Green kernel of an infinite graph"Monatshefte fur Math.. 138. 225-237 (2003)

[Publications] S.Dragomir, T.Ichiyama, H.Urakawa: "Yang-Mills theory and conjugate connections"Differ.Geom.Its Appl.. 18. 229-238 (2003)

[Publications] N.Matsuura, H.Urakawa: "Discrete improper affine spheres"J.Geometry Physics. 45. 164-183 (2003)

[Publications] H.Urakawa: "Yang-Mills theory over compact symplectic manifolds"Annals Global Analysis Geometry. (印刷中).

[Publications] H.Urakawa: "The heat kernel and the Green kernel of an infinite graph"Contemporary Math.Amer.Math.Soc.. (印刷中). (2004)

[Publications] H.Urakawa: "The Dirichet eigenvalue problem, the finite element method and graph theory"Contemporary Math.Amer.Math.Soc.. (印刷中). (2004)

[Publications] A.Hora: "Scaling limit for Gibbs states of Johnson graphs and resulting Meixner classes"Infin.Dimen.Anal.Quantum Probab.Rel.Top.. 6. 139-143 (2003)

[Publications] A.Hora: "A noncommutative version of Kerov's Gaussian limit for the Plancherel measure of the symmetric group"Lecture Notes in Math.. 1815. 77-88 (2003)

[Publications] T.Arimitsu: "Quantum stochastic differential equations in view of non-equilibrium thermo field dynamics"Fundamental Aspects of Quantum Physics(L.Accardi and S.Tasaki, Eds.). 24-47 (2003)

[Publications] T.Arimitsu, N.Arimitsu: "Analysis of accelerations in turbulence based on generalized statistics"Condensed Matter Phys.. 6. 85-92 (2003)

[Publications] P.Jizba, T.Arimitsu: "The world according to Renyi : Thermodynamics of fractal systems"Annals of Physics. (印刷中).

[Publications] T.Arimitsu, N.Arimitsu: "Multifractal analysis of fluid particle accelerations in turbulence"Physica D. (印刷中). (2003)

[Publications] A.E.Kobryn, T.Hayashi, T.Arimitsu: "Quantum stochastic differential equations for Boson and Fermion systems ---Method of non-equilibrium thermo field dynamics"Annals of Physics. 308. 395-446 (2003)

[Publications] A.E.Kobryn, T.Hayashi, T.Arimitsu: "Quantum stochastic processed : Boson and Fermion Brownian motion"Condensed Matter Phys.. 6. 367-646 (2003)

[Publications] P.Jizba, T.Arimitsu: "On observability of Renyi's entropy"Phys.Rev.E. (印刷中).

[Publications] A.E.Kobryn, T.Hayashi, T.Arimitsu: "Boson and Fermion Brownian motion"J.Phys.Soc.Japan. 72. 58-61 (2003)

[Publications] N.Obata, K.Saito: "Cauchy processes and the Levy Laplacian"Quantum Probability and White Noise Analysis. 14. 360-373 (2003)

[Publications] K.Nishi, K.Saito, A.H.Tsoi: "Fractional Browian motions and the Levy Laplacian"QUantum Information. 5(印刷中). (2004)

[Publications] K.Nishi, K.Saito: "An infinite dimensional stochastic process and the Levy Laplacian acting on WND-valued functions"Quantum Information and Complexity. (印刷中). (2004)

[Publications] 明出伊類似, 尾畑伸明: "量子確率論の基礎"牧野書店. 296 (2003)

[Publications] F.Hiai, H.Kosaki: "Means of Hilbert Space Operators"Springer-Verlag. 148 (2003)

尾畑伸明東北大学, 大学院・情報科学研究科, 教授 (10169360)