Stochastic systems with complex interactions and random environments

该项目研究了几种具有复杂相互作用的随机过程，在某些情况下也具有非均匀或随机环境：相互作用粒子系统，聚合物模型和随机介质中的随机运动。目标是描述典型的大规模行为，并量化与典型行为的偏差。在聚合物模型中，目标是解决长期存在的关于描述分子链和自由能波动数量级的缩放指数的开放性问题。对于一类非对称零距过程，本项目提出证明确认kpz型行为的时空相关和电流波动的标度性质。对于在随机环境中运动的单个粒子，本项目研究了大偏差，特别是当淬灭和平均大偏差的速率函数重合时的问题。对于在动态变化的环境中运动的大量粒子，目标是证明电流的分布极限定理。该项目研究描述不规则环境中粒子的复杂相互作用和运动的数学模型。这些数学系统结合随机性来模拟不规则性和不可预测性。目标是发现控制这些系统的一般数学原理。关键的一点是，这些系统在微观和宏观尺度上表现得截然不同。因此，了解小规模相互作用和运动的不同规则如何导致不同的大规模系统行为是很重要的。这种数学研究可以阐明的现实世界现象包括高速公路上车辆的运动、通过通信网络的数据包、管道中的流体颗粒、流体在多孔介质中传播的润湿过渡、种群中个体之间传播的流行病，或者流体中聚合物链的波动。从长远来看，理解这些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。数学家、自然科学家、社会科学家和工程师都在密切地、同时地研究这类模型。

This project studies several classes of stochastic processes that possess complicated interactions and in some cases also inhomogeneous or random environments: interacting particle systems, polymer models, and random motion in a random medium.The goal is to describe typical large scale behavior and to quantify deviations from the typical behavior.In polymer models the goal is to settle long-standing open problems on scaling exponents that describe the order of magnitude of the fluctuations of the molecule chain and the free energy.For a class of asymmetric zero range processes this project proposes to prove scaling properties of space-time correlations and current fluctuations that confirm KPZ-type behavior.For a single particle moving in a random environment this project studies large deviations, especially the question of when the rate functions for quenched and averaged large deviations coincide.For a large collection of particles moving in a dynamically evolving environment the goal is to prove distributional limit theorems for the current.This project investigates mathematical models that describe complex interactions and motion of particles in an irregularenvironment. These mathematical systems incorporate randomnessto model irregularity and unpredictability.The goal is to discover general mathematical principlesthat govern such systems. A key point is that thesesystems appear quite different at microscopic and macroscopic scales. So it is important to understand how different rules for small-scale interactions and motions lead to different large-scale systemwide behavior.Real-world phenomena that such mathematical studies can illuminate include the motion of vehicles on a freeway, packets making their way through a communication network, fluid particles in a tube, wetting transitions where fluid spreads in a porous medium, epidemics advancing among individuals in a population, or the fluctuations of a polymer chain in a fluid.Over the longer term understanding these complex interactions has profound implications for science and engineering and thereby for society. Models of the kind described in the proposal are intensely and concurrently studied by mathematicians, natural scientists, social scientists, and engineers.