Mathematical Sciences: Complex Stochastic Systems

9504323：Kurtz/Ney摘要。 研究项目将在 马尔可夫过程，随机分析，无限长链 记忆、大偏差理论、分支过程和随机 控制 马尔可夫过程的工作将集中在测量值 与群体遗传学和其他群体模型有关的过程。 关于大偏差的工作将包括进一步的理论发展和模拟应用。 将研究电子-光子级联和相关物理过程的分支过程模型，并进一步应用大偏差方法。 一般随机控制问题的理论发展将包括近似方法和最优奇异控制的存在定理。 将研究马尔可夫过程的新型随机方程。 马尔可夫过程是一种随机过程，其未来行为仅通过现在依赖于过去。 如果你知道一个具有马尔可夫性质的系统的当前“状态”，那么你预测未来的能力就不会得到关于过去的额外信息的帮助。 马尔可夫过程在群体遗传学的研究中起着重要的作用，并在更普遍的变化群体的描述，并将在此背景下进行研究。 马尔可夫过程作为宇宙射线等物理现象的模型也很重要，我们将对此进行研究。 要考虑的问题包括控制、近似和极端行为。 另一个研究方向涉及不具有“马尔可夫性质”，但具有特殊类型的长距离记忆的过程。

9504323: Kurtz/Ney Abstract. Research programs will be carried out in the areas of Markov processes, stochastic analysis, chains with infinite memory, large deviation theory, branching processes, and stochastic control. Work on Markov processes will focus on measure-valued processes related to population genetics and other population models. Work on large deviations will include both further theoretical developments and applications to simulation. Branching process models for electron-photon cascades and related physical processes will be studied with additional applications of large deviation methods. Theoretical developments for general stochastic control problems will include approximation methodology and existence theorems for optimal singular controls. New types of stochastic equations for Markov processes will be studied. Markov processes are random processes whose future behavior depends on the past only through the present. If you know the current "state" of a system having the Markov property, your ability to predict the future is not helped by additional information about the past. Markov processes play an important role in the study of population genetics and in descriptions of changing populations more generally, and will be studied in this context. Markov processes are also important as models of physical phenomena such as cosmic rays, which will be investigated. Problems to be considered include control, approximation, and extreme behavior. An additional direction of research involves processes which fail to have the "Markov property", but which have special kinds of long range memory.