Probabilistic Research of Large Scale Interacting Systems

大尺度相互作用系统一般意味着引入数学模型，从微观层面解释和理解宏观层面的物理现象。这类系统通常表现出非常复杂的特征。本研究的目的是统一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，本研究在2002年的第一年组织了“大规模相互作用系统的随机分析”国际研讨会。这对寻找研究方向起到了重要的作用。研究对象广泛、涉及面广。让我们具体说明其中一些：晶格界面模型、自由边界推导、问题、大偏差、准winterbottom形状、Widom-Rowlinson模型中的相结构和相分离曲线、相互作用布朗粒子、随机矩阵、抛物型Anderson模型、随机介质中的聚合物、非交叉随机漫步、流行病模型和相共存、演化奇异极限与固定、燃烧模型与奇异项偏微分方程、两相障碍模型、具有奇异项的随机偏微分方程，限制维纳测度的分部积分公式，贝塞尔过程的维纳积分。研究结果表明：从界面模型或双组分系统出发，推导演化变分不等式、自由边界问题等具有奇异点的微分方程；从界面模型或双组分系统出发，寻找随机漫步或聚合物模型与界面模型的相似性；从不同角度研究相互作用的布朗粒子；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.