Stochastic systems with complex interactions and random environments

这个项目研究了三类随机过程：相互作用的粒子系统、界面模型和随机介质中的随机运动。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。具体的研究对象包括排除过程和零程过程中电流的波动和大偏差，随机环境中排除过程的行为，随机平均过程中的高度波动，以及随机环境中随机游动的熄灭的中心极限定理。一个需要澄清的重要现象是，随着零程过程参数的变化，不同普适类之间的过渡。该项目研究描述粒子在不规则环境中的复杂相互作用和运动的数学模型。这些数学系统结合了随机性来模拟不规则性和不可预测性。其目标是挖掘出支配这类系统的一般数学定律，而不考虑不那么重要的细节。一个关键点是，这些系统在微观和宏观尺度上看起来截然不同。因此，了解小尺度相互作用和运动的不同规则如何导致不同的大尺度系统行为是很重要的。这类数学研究可以解释的真实世界现象包括高速公路上的车辆运动、通过通信网络的信息包、管道中的流体颗粒、流体在多孔介质中传播时的润湿过渡，或者流行病在果园中传播。从长远来看，理解复杂的相互作用对科学和工程，进而对社会都有广泛的影响。数学家、自然科学家、社会科学家和工程师正在对提案中描述的那种模型进行深入而同时的研究。

This project studies three categories of stochastic processes: interacting particle systems, interface 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. Examples of particular objects of study are fluctuations and large deviations of the current in the exclusion process and the zero range process, behavior of the exclusion process in a random environment, height fluctuations in the random average process, and quenched central limit theorems for random walk in a random environment. An important phenomenon to clarify is the transition between different universality classes as the parameters of the zero range process are varied. This project investigates mathematical models that describe complex interactions and motion of particles in an irregular environment. These mathematical systems incorporate randomness to model irregularity and unpredictability. The goal is to unearth general mathematical laws that govern such systems irrespective of less important details. A key point is that these systems appear quite different at microscopic and macroscopic scales. So it is important to understand howdifferent 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, or epidemics advancing through an orchard of trees. Over the longer term understanding complex interactions has wide 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.