Synthetic Research of Probability Theory

概率论综合研究

• 批准号: 17204009
• 负责人: SUGITA Hiroshi
• 金额: $ 30.53万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17204009/
• 关键词: Probability theor; Stochastic analysis; Stochastic Process; Limit theorem; Ereodic theory; Monte Carlo method; 確率過程論; 数理ファイナンス; マリアヴァン解析; ランダム分割

Our purpose is to contribute to Japanese society of probability theory by the total sum of research results of our investigators, who are experts of each field, in order to develop and make use of the traversality that probability theory originally has.We sponsored or co-sponsored 4 international conferences (Probability and Number theory, Large Scale Interaction Systems, 2 conferences on Mathematical Finance), 26 domestic symposia, in which we financially supported 285 participants including 13 from overseas.We got many results. Among them, we here present 3 results obtained by the head investigator. (1) By formulating the Monte Carlo method as stochastic game (gambling), and applying Kolmogorov's random number theory, we revealed the essential problem of sampling in the Monte Carlo method (2) The probability of two monic polynomials over any finite fields to be coprime is computed by an extended ergodic theorem in the adelic completion of the ring of polynomials. (3) We considered a randomized digamma function, and investigated the central limit theorem for them, whose results can be regarded as the fluctuated potential caused by randomly located electric charges on the half line.

我们的目的是通过总结各领域研究人员的研究成果，为日本概率论社会做出贡献，以发展和利用概率论原本具有的传播性。我们主办或联合主办了4次国际会议(概率与数论，大规模交互系统，2次数学金融会议)，26次国内研讨会，我们资助了285名与会者，其中13名来自国外。我们获得了许多成果。其中，我们在这里介绍了首席调查员获得的3项成果。(1)通过将蒙特卡罗方法描述为随机博弈(赌博)，并应用Kolmogorov随机数理论，揭示了蒙特卡罗方法中抽样的本质问题。(2)利用多项式环的Adelic完备性中的一个推广的遍历定理，计算了任意有限域上的两个一次多项式互素的概率。(3)考虑了一个随机化的Digamma函数，研究了它的中心极限定理，其结果可以看作是由半直线上随机分布的电荷引起的位势的起伏。

项目成果

Random walk on the incipient infinite cluster on trees

• DOI: 10.1215/ijm/1258059469
• 发表时间: 2005-03
• 期刊: Illinois Journal of Mathematics
• 影响因子: 0.6
• 作者: M. Barlow;T. Kumagai

Reflecting Ornstein-Uhlenbeck processes on pinned path spaces.

在固定路径空间上反映 Ornstein-Uhlenbeck 过程。

• 发表时间: 2008
• 期刊: RIMS Kokyuroku Bessatsu B6
• 作者: M. Hino;H. Uchida
• 通讯作者: H. Uchida

Branching Brownian motions on Riemannian manifolds : Expectation of the number of branches hitting closed sets.

黎曼流形上的分支布朗运动：命中闭集的分支数量的期望。

• 期刊: Potential Analysis (掲載確定)
• 作者: Hideyuki Ishi;Takaaki Nomura;M.Takeda
• 通讯作者: M.Takeda

On the Jacobi field approach to stochastic oscillatory integrals with quadratic phase function

二次相位函数随机振荡积分的雅可比场方法

• 发表时间: 2007
• 期刊: Kyushu J. Math. 61
• 作者: Taniguchi;Setsuo
• 通讯作者: Setsuo

Renormalized Rauzy inductions

重正化 Rauzy 归纳法

• 发表时间: 2007
• 期刊: Advanced Studies in Pure Math. 49
• 作者: Morita;Takehiko
• 通讯作者: Takehiko