Research in Stochastic Processes

0404952马库斯教授马库斯和罗森计划完成他们的书，“马尔可夫过程，高斯过程和当地时间”，其中包括他们的研究结果在过去的十六年中使用同构定理研究当地时间的对称马尔可夫过程采用发达的高斯过程理论。他们预计，这本书将刺激新的研究，这将导致在随机过程和可能的物理学的重大见解。他们还打算继续他们的研究集中在第一射线骑士定理，一个经典的同构定理的局部时间的布朗运动，他们还没有能够扩大，并考虑一个问题提出的西曼齐克在量子场论，涉及自相交局部时间。马库斯教授计划继续他的研究连续性和有界性的随机卷积相对于无限可分过程。教授罗森打算研究大偏差和指数可积的马尔可夫过程的泛函有关的交叉点。他还打算研究精细性质的几何简单随机游走在两个层面。 Marcus和罗森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，即使这些过程是随机的，它们也包含一些基本的内部结构，当理解这些结构时，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，季节也在变化。这里有一个基本的问题：是平均气温正在变暖，还是我们似乎正在目睹的变暖只是一个基本稳定的天气模式的短期波动？这项研究并不试图回答这个具体问题。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诸如此类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.