Research in Stochastic Processes

[404952] Marcus教授和Rosen教授计划完成他们的著作《马尔可夫过程，高斯过程和局部时间》，其中包括他们在过去16年中利用同构定理研究对称马尔可夫过程局部时间的研究成果，并采用了发达的高斯过程理论。他们预计这本书将激发新的研究，这将导致对随机过程和可能的物理学的重大见解。他们还打算继续研究第一雷-奈特定理，这是布朗运动局部时间的一个经典同构定理，他们无法推广，并考虑赛门齐克在量子场论中提出的一个涉及自交局部时间的问题。Marcus教授计划继续他的研究关于无限可分过程的随机卷积的连续性和有界性。Rosen教授打算研究与交点相关的马尔可夫过程泛函的大偏差和指数可积性。他还打算研究二维简单随机漫步的精细几何性质。Marcus教授和Rosen教授的研究旨在理解随机过程的结构。随机过程是随机现象随时间演化的模型。然而，尽管这些过程是随机的，但它们包含一些基本的内部结构，当我们理解这些结构时，它们在某种意义上是可预测的。全球变暖就是一个很好的例子。气温每天都在变化，每个季节都在变化。有一个基本的问题：是平均气温在变暖，还是我们所看到的变暖只是一个基本稳定的天气模式的短期波动？这项基金支持的研究并不试图回答这个具体问题。相反，它是对随机结构基本属性的研究，它可能会提供更有效地处理像这个这样的重要问题的工具。

0404952Marcus Professors Marcus and Rosen plan to complete their book, `Markov Processes, Gaussian Processes and Local Times', which includes the results of their research over the last sixteen years using isomorphism theorems to study local times of symmetric Markov processes by employing the well developed theory of Gaussian processes. They anticipate that the book will stimulate new research, which should lead to significant insights in stochastic processes and possibly physics. They also intend to continue their research concentrating on the First Ray-Knight Theorem, the one classical isomorphism theorem for the local times of Brownian motion that they have not been able to extend, and to consider a problem raised by Symanzik in quantum field theory, involving self-intersection local times. Professor Marcus plans to continue his research on the continuity and boundedness of stochastic convolutions with respect to infinitely divisible processes. Professor Rosen intends to study large deviations and exponential integrability for functionals of Markov processes related to intersections. He also intends to study fine properties of the geometry of the simple random walk in two dimensions. The research of Professors Marcus and Rosen is aimed at understanding the structure of stochastic processes. Stochastic processes are models for the evolution of random phenomena in time. However, even though the processes are random they contain some fundamental inner structure which when understood makes them in some sense predictable. Global warming is a good example. The temperature varies, day by day and season by season. There is a fundamental question: Is the average temperature getting warmer, or is the warming we seem to be witnessing simply a short term fluctuation of a basically stable weather pattern? The research supported by this grant does not attempt to answer this specific question. It is instead a study of the fundamental properties of random structures, which may give the tools to more effectively deal with important questions such as this one.