Complex Stochastic Systems

9971571库尔茨研究计划将在随机分析，测量值过程，随机滤波和控制，大偏差和空间点过程等领域进行。在随机分析领域的工作将集中在随机偏微分方程，随机方程的模拟方案的分析，以及随机方程和鞅问题之间的关系。推导并分析了非标准观测结构的滤波方程。以前的工作鞅的特征定期控制问题将扩展到奇异和部分可观测控制问题。基于非线性半群收敛的马尔可夫过程大偏差的研究方法将被推广到新的模型类。将研究马尔可夫生灭过程的平稳分布和由泊松随机测度驱动的随机方程的解。这项工作将集中在空间遍历性和中心极限定理与应用程序的参数估计铭记。随机过程的研究关注的是由“随机”或“机会”机制控制的自然现象的数学描述。这种现象的数学模型可能试图描述时间、空间或两者的变化。要进行的研究是关注开发方法来指定这些数学模型，近似复杂的模型由简单的，从损坏的或“嘈杂”的观察获得有关系统的真实状态的信息，并确定如何影响或“控制”的演变的模型和它们所代表的现象。特别地，滤波方程提供了一种通过使用未观察到的现象与可以观察到的量之间的关系的数学模型来处理关于不能直接和/或准确观察到的现象的信息的方法。将被考虑的情况的例子包括模型，其中所观察到的信息是空间分布的，例如可以从卫星观测和通信网络的模型，其中一个试图获得关于网络的状态的信息，从观察在网络中的几个点。

9971571Kurtz Research programs will be carried out in the areas of stochastic analysis, measure-valued processes, stochastic filtering and control, large deviations, and spatial point processes. Work in the area of stochastic analysis will focus on stochastic partial differential equations, the analysis of simulation schemes for stochastic equations, and the relationship between stochastic equations and martingale problems. Filtering equations for nonstandard observation structures will be derived and analyzed. Previous work on the martingale characterization of regular control problems will be extended to singular and partially observed control problems. Methods for the study of large deviations for Markov processes based on convergence of nonlinear semigroups will be extended to new classes of models. Spatial point processes characterized as stationary distributions of Markov birth and death processes and as solutions of stochastic equations driven by Poisson random measures will be studied. The work will focus on spatial ergodicity and central limit theorems with applications to parameter estimation in mind. The study of stochastic processes is concerned with mathematical descriptions of natural phenomena governed by "random" or "chance" mechanisms. Mathematical models of such phenomena may attempt to describe variation in time, in space, or both. The research to be performed is concerned with developing methods for specifying these mathematical models, approximating complex models by simpler ones, obtaining information about the true state of a system from corrupted or "noisy" observations, and determining how to influence or "control" the evolution of the models and the phenomena they represent. In particular, filtering equations provide a method of processing information about phenomena that cannot be directly and/or accurately observed, by using a mathematical model for the relationship between the unobserved phenomena and quantities that can be observed. Examples of the kind of situations that will be considered include models in which the observed information is spatially distributed such as might be obtained from satellite observations and models of communications networks in which one attempts to obtain information about the state of the network from observations at a few points in the network.