Synthetic Research of Probability Theory

概率论综合研究

• 批准号: 11304003
• 负责人: FUNAKI Tadahisa
• 金额: $ 18.69万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11304003/
• 关键词: Stochastic analysis; Markov processes; Hydrodynamic limit; Random matrices; Stochastic control; Ergodic theory; Infinite particles' system; Dirichlet form

This research was accomplished by 25 members under strong helps from many researchers in probability theory and related fields. During three years of the research period, 27 meetings were organized and 21 researchers were invited from abroad. A lot of research products were obtained in broad area:1. Related to the basic theory in probability theory, extension of the inverse arcsine law to one dimensional diffusion processes, properties of Brownian motion/heat kernel/Green function on several spaces, Markov processes and Dirichlet form, infinite dimensional stochastic analysis, stationary phase method and asymptotic theory and others were discussed.2. As applications of probability theory, mathematical physics such as analysis of phase separating surface arising under phase transitions, derivation of free boundary problem, motion of interacting Brownian hard balls, scaling limit of percolation cluster, random matrix, stochastic processes on fractals, problem of risk sensitive stochastic control, ergodid theory especially law of large numbers for ψ-mixing random variables, numerical calculation for stochastic differential equations and nonlinear diffusion equations, asymptotic expansion and applications of Malliavin calculus in mathematical statistics, problems related to differential geometry like collapse of manifolds.Concrete explanations on each research result can be found in the booklet of the research report in details. As is stated above, under the project of this research, many foreign researchers were invited and it was very fruitful and extremely important for the development of the probability theory in Japan in future.

本研究由25名成员在众多概率论及相关领域研究人员的大力帮助下完成。在三年的研究期间，组织了27次会议，邀请了21名国外研究人员。在广泛的领域取得了大量的研究成果：1.结合概率论中的基本理论，讨论了反正弦律在一维扩散过程中的推广，几种空间上布朗运动/热核/绿色函数的性质，马尔可夫过程与Dirichlet形式，无穷维随机分析，稳相方法与渐近理论等.作为概率论的应用，数学物理如相变下的相分离面分析，自由边界问题的推导，相互作用的布朗硬球的运动，渗流集团的标度极限，随机矩阵，分形上的随机过程，风险敏感随机控制问题，遍历理论特别是混合随机变量的大数定律，随机微分方程和非线性扩散方程的数值计算，Malliavin微积分在数理统计中的渐近展开和应用，微分几何的相关问题，如流形的坍缩，每一项研究成果的具体解释都可以在研究报告的小册子中找到。如上所述，在这项研究的项目下，许多外国研究人员被邀请，这是非常富有成效的，对日本概率论的发展非常重要。

项目成果

長井英生: "確率微分方程式"共立出版. 227 (1999)

永井秀夫：“随机微分方程”Kyoritsu Shuppan 227 (1999)。

T.Hara: "The scaling limit of the incipient infinite cluster in high-dimensional percolation.II.Integrated super-Brownian excursion."J.Math.Phys.. 41. 1244-1293 (2000)

T.Hara：“高维渗透中初始无限星团的标度极限。II.积分超布朗偏移。”J.Math.Phys.. 41. 1244-1293 (2000)

Y.Ogura(+富崎松代): "Superposition of diffusion processes-Feller Property-."Trends in Probability and related analysis.. 113-128 (1999)

Y.Ogura（+Matsuyo Tomizaki）：“扩散过程的叠加 - Feller Property -”。概率趋势和相关分析.. 113-128 (1999)

I.Shigekawa: "The domain of a generator and the intertwining property."Stochastics in Finite and Infinite Dimensions,. 401-410 (2000)

I.Shigekawa：“生成器的域和交织的属性。”有限和无限维度中的随机变量。

N.Yoshida: "Application of log-Sobolev inequalitv to the stochastic dynamics of unbounded spin systems on the lattice."J. Funct.Anal.. 173. 74-102 (2000)

N.Yoshida：“对数索博列夫不等式在晶格上无界自旋系统的随机动力学中的应用。”J.