Stochastic Systems with Complex Interactions and Random Environments

这个项目是对数学模型的基础研究，该模型描述了复杂的相互作用、生长和在不规则环境中的运动。这些系统在小尺度和大尺度上表现出非常不同的特征，理解小尺度演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、通信网络中的分组、管道中的流体颗粒、流体在多孔介质中传播时的润湿转变、流行病在人群中的传播，或者流体中聚合物链的波动。这个项目的目标是发现支配这类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同时对这些系统进行密集和同时的研究。从长远来看，了解这些和其他复杂的相互作用对科学和工程，从而对社会具有深远的影响。该项目通过参与研究来培训博士生。该项目研究随机环境中的随机路径和随机增长模型。其目标是描述典型的大规模行为，并量化与典型行为的偏差。重点是找到适用于具有共同基本特征的模型类别的普遍原则。当前提议的主要方向是在这些模型中发现和开发新的数学结构。这些结构是被称为余循环和马尔可夫过程的类梯度函数。这些物体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。这个项目试图通过变分公式来刻画极限对象的特征。求解变分公式的余循环定义了模型的不变版本，并可用于研究波动。首要目标是建立模型的通用属性，超越有限的显式可解情况集。

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.