Infinite Dimensional Stochastic Processes and the Information Analysis

无限维随机过程与信息分析

• 批准号: 15540141
• 负责人: SAITO Kimiaki
• 金额: $ 1.22万
• 依托单位: Meijo University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540141/
• 关键词: Infinite dimensional stochastic process; Levy Laplacian; Feynman path integral; Quantum stochastic process; Information Analysis; White Noise; Fractional Brownian motion; Schroedinger equation

We deeply appreciate the grant for scientific research (term : academic years 2003, 2004) from JSPS. In this research, we considered roles as an information analysis in researching the infinite dimensional stochastic analysis jointly from major fields probability theory, analysis, variational geometry, number theory and computer science.Main results which we obtained are the following : 1) By changing the state space, we constructed an infinite dimensional Wiener process associated with the Levy Laplacian, and moreover we can extend this Laplacian operator to an operator on a space of operator- valued white noise functionals to get 2). 2) We also constructed a quantum stochastic process generated by the quantum Levy Laplacian. Based on this result a quantum information analysis associated the Levy Laplacian can be expanded to discuss the quantum communication theory 3) By using an infinite sequence of independent Brownian motions we constructed an infinite dimensional stochastic proces … More s generated by a sum of the Levy Laplacian. 4) Introducing a compensated stochastic process as a difference of two independent Levy processes, we construct a Flock space based on the Levy Laplacian. This method can be applied to construct a quantum stochastic process. 5) The method of calculating the Feynman path integrals in quantum field theory can be formulated using white noise distribution theory, and the Levy Laplacian' and the Volterra Laplacian appear in the Schroedinger type equation. 6) 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.By the above results our joint research with Professor Kuo of Louisiana State University in USA was developed research of the quantum probability theory approached by white noise operator theory. Moreover we had results on entropy of infinite sequences, which is connecting to the quantum entropy.Proceeding with the above research, we organized a seminar every week and discussed each theme in the infinite dimensional stochastic analysis and information analysis between co-researchers. We gave talks on results in this research at the international conferences in Tunisia and in Italy. Through talks and discussions in several international conferences, many researchers were interested in our results and we could start new joint works with participants in the conferences. In the international conference in Italy, I had to attend at the committee of International Association " Quantum Probability and Infinite Dimensional Analysis" as a member. We expect many developments on this research with joint researchers in near future. Less

我们非常感谢JSP的科学研究赠款（学期：学年：2003年，2004年）。在这项研究中，我们将角色视为信息分析在研究无限尺寸随机分析中，共同从主要领域概率理论，分析，变异几何，数量理论和计算机科学。我们获得的结果是以下内容：1）通过更改状态空间，我们可以通过更改状态空间，我们构建了一个无限的wiener dimensientian foramian上的操作员，我们可以扩展该操作员，并扩展了一定的运算范围，我们可以扩展一定的laplac和更多的操作。运算符值的白噪声功能可获得2）。 2）我们还构建了由量子征收拉普拉斯（Laplacian）产生的量子随机过程。基于此结果，可以扩展量子信息分析相关的量子信息分析，以讨论量子通信理论3）通过使用独立的布朗尼动作的无限序列，我们构建了一个无限的维度随机过程……更多由Laplacian产生的更多S。 4）引入一个补偿随机过程作为两个独立征税过程的差异，我们基于征税拉普拉斯（Laplacian）构建了一个羊群。该方法可以应用于构建量子随机过程。 5）可以使用白噪声分布理论来制定量子场理论中Feynman路径积分的方法，而Laplacian'和Volterra Laplacian出现在Schroedinger类型方程中。 6）我们获得了征费的拉普拉斯和无限尺寸分数Ornstein-Uhlenbeck工艺之间的关系。这种关系很重要，必须基于征费拉普拉斯的随机分析作为数学金融。此外，我们可以扩展这一结果，以获得拉普拉斯和一般无限的无限尺寸Ornstein-Ohlenbeck过程。通过上述结果，我们与美国路易斯安那州立大学的Kuo教授的联合研究是对白噪声操作员理论对量子概率理论进行的研究的研究。此外，我们在无限序列的熵上得到了结果，该序列正在连接到量子熵。与上述研究相关，我们每周组织一个开创性，并在共同研究者之间的无限维度随机分析和信息分析中讨论了每个主题。我们在突尼斯和意大利的国际会议上进行了有关这项研究结果的讨论。通过在几个国际会议上的谈判和讨论，许多研究人员对我们的结果感兴趣，我们可以与会议的参与者开始新的联合合作。在意大利的国际会议上，我必须参加国际协会委员会作为成员“量子概率和无限分析”。我们希望在不久的将来与联合研究人员有关这项研究的许多发展。较少的

项目成果

Recent progress on the white noise approach to the Levy Laplacian

Levy Laplacian 的白噪声方法的最新进展

• 发表时间: 2004
• 期刊: Quantum Information and Complexity Vol.1
• 作者: K.Saito;K.Nishi;K.Sakabe;H.-H.Kuo
• 通讯作者: H.-H.Kuo

An infinite dimensional Laplacian acting on some class of Levy white noise functionals

作用于某类 Levy 白噪声泛函的无限维拉普拉斯算子

• 发表时间: 2005
• 期刊: Proc.Quantum Information 2003 (L. Accardi eds.) (World Scientific) (to appear)
• 作者: K.Saito;K.Saito;K.Saito;K.Saito
• 通讯作者: K.Saito

Fractional Brownian motions and the Levy Laplacian

分数布朗运动和 Levy Laplacian

• 发表时间: 2005
• 期刊: Quantum Information V, World Scientific (to appear)
• 作者: K.Nishi;K.Saito;A.H.Tsoi
• 通讯作者: A.H.Tsoi

K.Nishi, K.Saito: "An infinite dimensional stochastic process and the Levy Laplacian acting on WND-valued functions"Quantum Information and Complexity. Vol.1(To appear). (2004)

K.Nishi、K.Saito：“无限维随机过程和作用于 WND 值函数的利维拉普拉斯算子”量子信息和复杂性。

K.Saito, K.Nishi, K.Sakabe: "Infinite dimensional Brownian motions and Laplacian operators in white noise analysis"岐阜高専紀要. 39号. 17-26 (2004)

K.Saito、K.Nishi、K.Sakabe：“白噪声分析中的无限维布朗运动和拉普拉斯算子”岐阜国立工业大学公报第 39 期。17-26（2004 年）。