Research in Stochastic Processes

随机过程研究

• 批准号: 0103253
• 负责人: Michael Marcus
• 金额: $ 18.6万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103253&HistoricalAwards=false
• 关键词: Research; Stochastic; Processes

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.

Marcus和罗森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。他们已经获得了许多有趣的结果，当地时间，他们已经发表了十几篇论文和一本专著。直到最近，他们的工作是基于一个同构定理Dynkin，这是很难证明和应用。在过去的两年里，这一切都改变了。与Eisenbaum、Kaspi和Shi教授一起，他们获得了与局部时间和高斯过程相关的新的、简单的同构，并大大简化和澄清了他们的早期工作。他们还获得了许多新的结果;最重要的是一个简化版本的雷定理的地方时间的扩散。他们将继续这项工作，以推广雷定理的范围，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考虑马尔可夫过程的其他属性，可以通过他们的本地时间进行研究。他们还计划通过将其与高斯混沌过程进行比较，将其结果扩展到更一般的强对称马尔可夫过程的连续可加泛函类。Marcus教授将继续研究无限可分移动平均过程的样本路径特性。这些过程是应用数学的基础。他们似乎有显着的光滑性，并表现得比类似定义的高斯过程。我们将对这一令人惊讶的观察结果进行调查。罗森教授计划研究一个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特别具有挑战性。他打算研究这个离散的问题有关它的一个连续的布朗运动的二维环面。这反过来又会导致对“晚点”的分析，这些点通过布朗路径的方法需要非常大量的时间。罗森教授还计划研究平面布朗运动路径上的无穷多个点。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，随时间演化的现象是以随机的方式进行的。例如道琼斯平均指数、全球变暖数据或卫星通信。特别重要的是一个进程占用特定值的时间量。这是研究的过程中的本地时间。在这个建议中，马尔可夫过程的局部时将通过相关的高斯过程进行研究。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，被认为是本质上无关的。马库斯教授和罗森教授已经表明，他们是密切相关的，并正在寻找一个统一的理论，这些重要的过程。