Collaborative Research: Research in Stochastic Processes

Marcus和Rosen教授将继续研究高斯过程与相关强对称马氏过程的局部时之间的关系。这是他们在2006年10月出版的剑桥大学出版社的书《马尔可夫过程、高斯过程和当地时间》中的主题。在写这本书的过程中，他们解决了许多问题，发现了许多新问题。他们特别感兴趣的是，基于样本路径性质，找到一个启发式的解释，即具有无限可除平方的高斯过程正是那些具有协方差的高斯过程，协方差是强对称马尔可夫过程的零势密度。他们还将继续使用高斯过程技术来发现本地时的新的样本路径性质的程序，例如关于马尔可夫过程的本地时连续模的中心极限定理。在继续探索高斯和马尔可夫过程之间的相互作用时，他们还将考虑具有凸协方差函数的高斯过程的连续模的非正态中心极限定理，作为获得本地时间类似性质的第一步。许多重要的现象，如天气模式或股票市场的行为，都是如此复杂，以至于研究它们的唯一方法是将它们视为随机或随机过程。研究随机过程的数学模型是为了深入了解它们所代表的物理现象。关于随机过程的微妙性质，是该过程在其可能的值上实现所花费的时间的量。此属性称为进程的本地时间。这一建议是为了继续研究对称马氏过程的局部时。

Professors Marcus and Rosen will continue their research on the relationship between Gaussian processes and the local times of related strongly symmetric Markov processes. This is the subject of their Cambridge University Press book, Markov Processes, Gaussian Processes and Local Times, which was published in October 2006. In writing this book they solved many problems and uncovered many new ones. They are particularly interested in finding a heuristic explanation, based on sample path properties, of the fact that Gaussian processes with infinitely divisible squares are precisely those Gaussian processes with covariance that is the zero potential density of a strongly symmetric Markov process. They will also continue their program of using Gaussian process techniques to discover new sample path properties of local times, such as central limit theorems for the moduli of continuity of local times of Markov processes. In continuing to explore the interplay between Gaussian and Markov processes they will also consider non-normal central limit theorems for the moduli of continuity of Gaussian processes with convex covariance functions as a first step towards obtaining similar properties for local times. Many important phenomena, such as weather patterns or the behavior of the stock market, are so complex that the only way to study them is to consider them as random, or stochastic, processes. Mathematical models of stochastic processes are studied to give insight into the physical phenomena that they represent. On subtle property of a stochastic process is the amount of time realizations of the process spend at the possible values that it can take. This property is called the local time of the process. This proposal is to continue research on the local times of symmetric Markov processes.