Markov Processes

这一提议在一定程度上是关于不同背景下的大偏差。在研究相互作用粒子极限的标度极限时，通常要证明大数定律。它们出现在不同的背景下，并具有相应的大偏差结果，这些结果是相关的。在适当的条件下，随机环境中的随机游动和扩散也被期望满足大数定律、中心极限定理和大偏差。大偏差理论与粘性为零的随机Hamilton-Jacobi-Bellman方程的齐次化理论有着密切的联系。我们预计将继续在这些地区进行调查。在复杂的交互系统中，噪声通常起到稳定力的作用，并引导系统沿着可预测的确定性路径前进。偏离这一行为的情况很少见。这个项目的一个目标是量化和测量给定的偏差有多罕见，如果这样的偏差应该发生，最有可能导致它的是什么？

This proposal is partly about large deviations in different contexts. In studying the scaling limits of interacting particle limits, one usually proves laws of large numbers. They arise in different contexts and have corresponding large deviation results that are related. Random walks and diffusions in a random environment are also expected, under suitable conditions, to satisfy law of large numbers, central limit theorem and large deviations. The large deviation theory has close connection to the theory of homogenization of random Hamilton-Jacobi-Bellman equations with vanishing viscosity. We expect to continue with our investigation in these areas. In complex interactive systems, noise often acts as a stabilizing force and guides the system along a predictable deterministic path. Deviations from this behavior are rare. One goal of this project is to quantify and measure how rare a given deviation is and if such a deviation should occur what is most likely to have caused it?