Probabilistic Research of Large Scale Interacting Systems

大尺度相互作用系统一般是指从微观层次解释和理解物理现象的数学模型。这样的系统通常表现出非常复杂的特征。本文的研究目的是结合随机分析和非线性偏微分方程理论的观点，从多个方面对这些模型进行深入的研究。为了实现这一目标，在本研究的第一年，2002年，国际研讨会“随机分析大规模相互作用系统”组织。这对寻找研究方向起到了重要作用。让我们具体说明其中的一些：晶格界面模型，自由边界的推导，问题，大偏差，Widom-Rowlinson模型中的拟Winterbottom形状，相结构和相分离曲线，相互作用布朗粒子，随机矩阵，抛物安德森模型，随机介质中的聚合物，非交叉随机游动，传染病模型和相共存，钉扎演化奇异极限，燃烧模型和带奇异项的偏微分方程、两相障碍物模型、带奇异项的随机偏微分方程（如δ函数）、限制Wiener测度的分部积分公式、Bessel过程的Wiener积分。作为研究小组中研究人员有效合作的研究成果，我们陈述：从界面模型或两组分体系出发，寻找相互作用随机游动或聚合物模型与界面模型的相似性，从几个不同的方面研究相互作用布朗粒子，研究相分离曲线的行为等。

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.