Research in Stochastic Processes

Marcus教授Marcus和Rosen计划完成他们的著作《马尔可夫过程、高斯过程和本地时间》，这本书包含了他们在过去16年中的研究结果，他们使用同构定理通过使用成熟的高斯过程理论来研究对称马尔可夫过程的本地时间。他们预计，这本书将刺激新的研究，这应该会导致对随机过程的重大见解，可能还包括物理学。他们还打算继续他们的研究，集中在第一雷-奈特定理，一个经典的布朗运动局部时的同构定理，并考虑Symanzik在量子场论中提出的涉及局部自交的问题。Marcus教授计划继续研究关于无限可分过程的随机卷积的连续性和有界性。Rosen教授打算研究与交集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游动几何的精细性质。Marcus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，尽管过程是随机的，但它们包含一些基本的内部结构，当被理解时，这些结构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，季节也在变化。有一个根本的问题：平均气温是在变暖，还是我们正在见证的变暖只是基本稳定的天气模式的短期波动？这笔赠款支持的研究并不试图回答这个特定的问题。相反，它是对随机结构的基本性质的研究，这可能会为更有效地处理像这样的重要问题提供工具。

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.