Markov Process

这个项目的重点是相互作用的粒子系统和它们的尺度限制。当我们有一个大的粒子系统通过局部相互作用相互作用时，当它开始远离平衡时，这样一个系统的演化提出了几个问题。密度通常是唯一的守恒量，通常有某种输运方程作为标度极限。叠加在其上的是单个粒子的运动，在相互作用以某种方式在尺度极限内平均之后，这通常是一个非齐次马尔可夫过程。在这方面，大数定律以及波动和对它们的大偏差是一些有趣的问题。此外，在极限描述中产生的系数是密度和（或）其他参数的函数，它们对参数的依赖性的规律性也很重要。有一大类物理过程，其行为规则是在单个单位的水平上规定的。这些规则涉及到个体单位之间的互动性质，也可能包含一些随机性。但我们需要在更大的尺度上对单位的集体行为做出预测。这个项目涉及从个体单位层面的相互作用模型中推导出集体行为规则。在物理科学中，这类问题已被成功研究的例子包括由构成流体的分子之间相互作用的规律推导而来的流体流动规律。社会科学中的类似问题，例如根据个人之间的经济交换模型预测宏观经济行为的问题，还没有得到充分解决。

0104343Varadhan The focus of this project is interacting particle systems and their scaling limits. When we have a large system of particles interacting through local interactions, the evolution of such a system when it starts far away from equilibrium poses several questions. Density being often the only conserved quantity, there is usually some sort of a transport equation appearing as the scaling limit. Superposed on it is the motion of individual particles, which is usually an inhomogeneous Markov process after the interactions are somehow averaged in the scaling limit. Laws of large numbers as well as fluctuations and large deviations from them are some of the interesting problems in this context. In addition the coefficients arising in the limiting descriptions are functions of density and or other parameters and the regularity of their dependence on the parameters is also of importance. There are a large class of physical processes where the rules of behavior are prescribed at the level of individual units. These rules concern the nature of the interaction between individual units that could involve some randomness as well. But one needs to make predictions of the collective behavior of the units at a much larger scale. This project deals with the derivation of the rules of collective behavior from models of interactions at the level of individual units. In the physical sciences examples of such problems that have been successfully studied include the rules of fluid flow that are derived from the laws of interaction between molecules that make up the fluid. Similar problems in the social sciences, for instance one of predicting macroeconomic behavior based on models of economic exchange between individuals, have not been adequately addressed.