Stochastic Systems with Complex Interactions and Random Environments

该项目是对描述不规则环境中复杂相互作用、生长和运动的数学模型的基础研究。这些系统在小尺度和大尺度上表现出非常不同的特征，了解小尺度进化的不同规则如何导致不同的大尺度系统行为是很重要的。数学研究可以阐明广泛的过程，包括车辆的运动，通信网络中的数据包，管道中的流体颗粒，流体在多孔介质中传播的润湿过渡，种群中传播的流行病，或流体中聚合物链的波动。这个项目的目标是发现控制这些系统的一般数学规律，数学家、自然科学家、社会科学家和工程师都在密切研究这些规律。从长远来看，理解这些和其他复杂的相互作用对科学和工程，从而对社会有着深远的影响。该项目包括通过参与研究来培养博士生。本课题研究随机环境中的随机路径和随机增长模型。目标是描述典型的大规模行为，并量化与典型行为的偏差。重点是寻找适用于具有共同基本特征的模型类的普遍原则。当前建议的主要方向是在这些模型中发现和利用新的数学结构。这些结构是类似梯度的函数，称为共环和马尔可夫过程。这些对象被选为描述这些模型的极限自由能和极限形状的变分公式的极值。本项目试图通过变分公式来表征极限对象的特征。解变分公式的环定义了模型的不变版本，可以用来研究波动。首要目标是建立模型的普遍属性，而不是局限于一组显式可解的情况。

This project is fundamental research on mathematical models that describe complex interactions, growth, and motion in an irregular environment. These systems show very different features at small scales and at large scales, and it is important to understand how different rules for small-scale evolution lead to different large-scale system-wide behavior. Mathematical studies can illuminate a wide range of processes, including the motion of vehicles, packets in a communication network, fluid particles in a tube, wetting transitions where fluid spreads in a porous medium, epidemics advancing in a population, or the fluctuations of a polymer chain in a fluid. The goal of this project is to discover general mathematical laws that govern such systems, which are intensely and concurrently studied by mathematicians, natural scientists, social scientists, and engineers. Over the long term, understanding these and other complex interactions has profound implications for science and engineering and thereby for society. The project involves the training of Ph.D. students through involvement in the research.This project studies random paths in random environments and random growth models. The goal is to describe typical large scale behavior and to quantify deviations from the typical behavior. The emphasis is on finding universal principles that apply to classes of models that share fundamental characteristics. The main direction of the current proposal is to find and exploit new mathematical structures in these models. These structures are gradient-like functions called cocycles and Markov processes. These objects are selected as extrema of variational formulas that describe the limiting free energies and limit shapes of these models. This project attempts to characterize features of the limiting objects through the variational formulas. The cocycles that solve the variational formulas define invariant versions of the models and can be used to study fluctuations. The overarching goal is to establish universal properties for models beyond the narrow set of explicitly solvable cases.