Research in Stochastic Processes

Marcus和Rosen教授多年来一直在研究强对称马氏过程和高斯过程的局部时之间的关系。他们获得了许多关于当地时间的有趣结果，并在十几篇论文和一本专著中发表了这些成果。直到最近，他们的工作都是基于Dykin的同构定理，这一定理很难证明和应用。在过去的两年里，这一切都发生了变化。与Eisenbaum、Kaspi和Shih教授一起，他们得到了与当地时间和高斯过程有关的新的、简单的同构，并极大地简化和澄清了他们早期的工作。他们还得到了许多新的结果；其中最重要的是关于扩散的局部时的Ray定理的简化版本。他们将继续这项工作来推广雷定理的范围，以考虑不连续过程的局部时间。他们将应用他们的新结果和技术来研究马尔可夫过程的其他性质，这些性质可以通过他们的当地时间来研究。他们还计划将他们的结果推广到更一般的强对称马尔可夫过程的连续可加泛函类别，将它们与高斯混沌过程进行比较。马库斯教授将继续研究无限可分移动平均过程的样本轨道性质。这些过程是应用数学的基础。它们似乎具有显著的平滑特性，并且比类似定义的高斯过程表现得更好。这一令人惊讶的观察结果将被调查。罗森教授计划研究一次简单的随机漫步访问有限图上的每个点所需的时间。二维晶格环面的情况尤其具有挑战性。他打算通过将这个离散问题与一个关于二维环面上布朗运动的连续问题联系起来来研究它。这反过来将导致对“后期点”的分析，这些点通过布朗路径来接近花费了不寻常的大量时间。罗森教授还计划研究平面布朗运动路径上无限多重点。这项研究涉及随机过程的基本性质，在处理随机现象的所有领域都有潜在的应用。一般来说，随时间演变的现象是以随机的方式演变的。例如道琼斯平均指数、全球变暖数据或与卫星的通信。特别重要的是进程占用特定值的时间量。这是根据过程的当地时间进行研究的。在这个方案中，马尔可夫过程的局部时间将通过相关的高斯过程来研究。直到最近，这两类重要的随机过程，马尔可夫过程和高斯过程，还被认为本质上是不相关的。马库斯和罗森教授已经证明，他们是密切相关的，并正在为这些重要的过程寻找统一的理论。

Professors Marcus and Rosen have been studying the relationship between the local times of strongly symmetric Markov processes and Gaussian processes for many years. They have obtained many interesting results about local times which they have published in more than a dozen papers and a monograph. Until recently their work was based on an isomorphism theorem of Dynkin, which is difficult to prove and to apply. In the last two years this has all changed. Together with Professors Eisenbaum, Kaspi and Shi, they have obtained new, simple isomorphisms relating local times and Gaussian processes and have greatly simplified and clarified their early work. They have also obtained many new results; the most significant is a simplified version of Ray's theorem on the local times of diffusions. They will continue this work to generalize the scope of Ray's theorem to consider local times of processes which are not continuous. They will apply their new results and techniques to consider other properties of Markov processes that can be studied through their local times. They also plan to extend their results to more general classes of continuous additive functionals of strongly symmetric Markov processes by comparing them to Gaussian chaos processes. Professor Marcus will continue his studies of sample path properties of infinitely divisible moving average processes. These process are fundamental in applied mathematics. They appear to have remarkable smoothness properties and to behave better than similarly defined Gaussian processes. This surprising observation will be investigated. Professor Rosen plans to study the time needed for a simple random walk to visit each point on a finite graph. The case of the two dimensional lattice torus is particularly challenging. He intends to study this discrete problem by relating it to a continuous one concerning Brownian motion on the two dimensional torus. This in turn will lead to the analysis of `late points', those points whose approach by the Brownian path takes an unusually large amount of time. Professor Rosen also plans to study points of infinite multiplicity on the path of planar Brownian motion. This research deals with fundamental properties of stochastic processes and has potential applications in all areas that deal with random phenomena. Generally speaking phenomena that evolve in time do so in a random fashion. Examples are the Dow Jones average, data on global warming or communication with satellites. Of particular importance is the amount of time that a process takes a specific value. This is studied in terms of the local time of the process. In this proposal the local times of Markov processes will be investigated by means of associated Gaussian processes. Until very recently these two important classes of stochastic processes, Markov processes and Gaussian processes, were considered to be essentially unrelated. Professors Marcus and Rosen have shown that they are intimately related and are searching for a unified theory for these important processes.