Markov Process

0104343 Varadhan该项目的重点是相互作用的粒子系统及其缩放限制。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，当这样一个系统远离平衡时，它的演化会引起几个问题。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。叠加在它上面的是单个粒子的运动，这通常是一个非齐次马尔可夫过程，在相互作用以某种方式在标度极限中平均之后。在这方面，大数定律以及涨落和与它们的大偏差是一些有趣的问题。此外，在极限描述中出现的系数是密度和/或其他参数的函数，并且它们对参数的依赖性的规律性也是重要的。有一大类物理过程的行为规则是规定在个别单位的水平。这些规则涉及个体单元之间的相互作用的性质，这些相互作用也可能涉及一些随机性。但是，我们需要在更大的尺度上预测这些单元的集体行为。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的规则。在物理科学中，已经成功研究的这类问题的例子包括从组成流体的分子之间的相互作用定律导出的流体流动规则。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，还没有得到充分的解决。

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.