Probabilistic Research of Large Scale Interacting Systems

大规模交互系统的概率研究

• 批准号: 14340029
• 负责人: FUNAKI Tadahisa
• 金额: $ 8.96万
• 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 (B)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14340029/
• 关键词: Interface model; Hydrodynamic limit; Stochastic differential equation; Brownian paricles' system; Free boundary problem; Large scale interacting system; Stochastic analysis; Random media; Anderson模型; 非交叉ランダムウォーク; ランダム行列

The large scale interacting systems generically mean mathematical models which are introduced to explain and understand physical phenomena at macroscopic level from microscopic one. Such systems exhibit very complicated feature in general. The purpose of this research is, unifying the viewpoints of stochastic analysis and theory of nonlinear partial differential equations, to study these models from several aspects deeply. To achieve the goal, in the first year 2002 of this research, an international symposium "Stochastic Analysis on Large Scale Interacting Systems" was organized. This played an important role to find the direction of the research.The objects of the research were expansive and of wide range. Let us state some of them concretely : lattice interface model, derivation-of free boundary, problem, large deviation, quasi-Winterbottom shape, phase structure and phase separation curve in Widom-Rowlinson model, interacting Brownian particles, random matrices, parabolic Anderson model, polymers in random media, non-crossing random walks, epidemic model and phase coexistence, evolutional singular limit with pinning, combustion model and partial differential equation with singular term, two-phase obstacle model, stochastic partial differential equations with singular term like delta-functions, integration by parts formula for restricted Wiener measure, Wiener integrals for Bessel processes.As research results based on effective cooperation of investigators in the group, we state : derivation of differential equations with singularities, like evolutionary variational inequality and free boundary problem, starting from interface model or two component system, finding similarity in interacting random walks or polymer models with interface model, study of interacting Brownian particles from several different aspects, studying the behavior of phase separation curves and others.

大规模相互作用的系统通常是指数学模型，这些模型是为了解释和理解显微镜下宏观级别的物理现象。这种系统通常表现出非常复杂的功能。这项研究的目的是统一随机分析的观点和非线性偏微分方程的理论，以深入研究这些模型。为了实现这一目标，在这项研究的2002年第一年中，组织了国际研讨会“大规模相互作用系统的随机分析”。这是找到研究方向的重要作用。研究的对象是广泛的范围。 Let us state some of them concretely : lattice interface model, derivation-of free boundary, problem, large deviation, quasi-Winterbottom shape, phase structure and phase separation curve in Widom-Rowlinson model, interacting Brownian particles, random matrices, parabolic Anderson model, polymers in random media, non-crossing random walks, epidemic model and phase coexistence, evolutional singular limit with pinning, combustion模型和部分差分方程，具有单一术语，两相障碍模型，具有奇异术语（如三角形）的随机部分差分方程，零件公式进行限制性措施的整合，贝塞尔过程的维也纳积分，是研究结果的研究结果，基于研究人员的有效合作和差异范围的范围，我们的范围范围，我们的范围范围，我们的范围差异，诸如范围的范围，诸如衍生界的范围，诸如衍生界的范围，诸如衍生范围的范围，诸如衍生范围，诸如衍生范围的范围，诸如衍生范围的范围，诸如衍生范围的范围，诸如衍生范围的范围，既定的界限，诸如界限，诸如衍生品的范围，诸如范围模型或两个组件系统，在与界面模型的相互作用随机步行或聚合物模型中找到相似性，研究来自几个不同方面的布朗颗粒的研究，研究相分离曲线和其他方面的行为。

项目成果

Functional central limit theorems for vicious walkers.

恶意步行者的功能中心极限定理。

• 发表时间: 2003
• 期刊: Stochastics and Stochastics Reports 8
• 作者: N.Konno;N.Masuda;Norio Konno;Makoto Katori;Makoto Katori;Makoto Katori;Taro Nagao;Makoto Katori
• 通讯作者: Makoto Katori

The Two-Phase Membrane Problem - an intersection-comparison approach to the regularity at branch points.

两相膜问题 - 分支点规律性的交叉比较方法。

• 期刊: Advances in Mathematics (発表予定)
• 作者: H.Shahgholian;G.S.Weiss
• 通讯作者: G.S.Weiss

Phase transition from the viewpoint of relaxation phenomena.

从弛豫现象的角度来看相变。

• 发表时间: 2003
• 期刊: Review Math. Phys. 15
• 作者: H.J.Choe;G.S.Weiss;N.Yashida
• 通讯作者: N.Yashida

Hydrodynarnic limit and nonlinear PDEs with singularities.

流体动力极限和具有奇点的非线性偏微分方程。

• 发表时间: 2006
• 期刊: Advanced Studies in Pure Mathematics
• 作者: T.Funaki;T.Funaki
• 通讯作者: T.Funaki

Infinite systems of non-colliding Brownian particles.

非碰撞布朗粒子的无限系统。

• 发表时间: 2004
• 期刊: Advanced studies in Pure Mathematics 39
• 作者: N.Konno;N.Masuda;Norio Konno;Makoto Katori
• 通讯作者: Makoto Katori