Collaborative Research: Research in Stochastic Processes

马库斯教授和罗森教授将继续他们对永久过程的研究。这些是正实值过程包括高斯过程的平方。然而，高斯过程是由正定对称协方差函数定义的，而在永久过程的定义中，不需要对对称性和正定性的限制。永久过程是缺失的一环，可以用来将Dynkin同构定理扩展为包含不具有对称势密度的马尔可夫过程的局部时间的同构定理。他们还计划将永久过程的研究扩展到永久领域，并发展新的同构定理，将它们与连续加性泛函和一般马尔可夫过程的交集局部时间联系起来。他们希望能够使用它们来获得连续加性泛函和交集局部时间的样本路径性质。我们生活中的重要现象，如天气、金融市场、投票模式和对敌人活动的探测，都是如此复杂，以至于它们只能被建模为随机事件，称为随机过程。然而，尽管这些事件是随机的，但它们都有一定的结构，通常是不同的，这使我们能够对它们的行为做出很好的预测，这样我们就可以利用它们或保护自己不受它们的影响。许多概率学家研究随机过程。我们的专长是当地时间这是衡量这些过程的结果以及哪些结果更可能或更不可能的标准。我们的动机是双重的。一个是美学，因为底层的数学是非常美的。另一个是实用的，为工程师和科学家提供工具，保护我们免受破坏性天气的影响，控制破坏性的市场波动，分析选民模式，保护我们免受敌人导弹的攻击……潜在应用的清单是无穷无尽的。采用最先进的数学技术的设备和技术有助于新兴产业的发展，并将有助于保持我国的竞争力

Professors Marcus and Rosen will continue their research on permanental processes. These are positive real valued processes that include processes that are the squares of Gaussian processes. However, whereas Gaussian processes are defined by a positive definite symmetric covariance function, in the definition of permanental processes, the restrictions to symmetry and positive definiteness are not required. Permanental process are the missing link that can be used to extend the Dynkin Isomorphism Theorem to an isomorphism theorem that contains the local times of Markov processes that do not have symmetric potential densities. They also plan to extend the study of permanental processes to permanental fields and to develop new isomorphism theorems that relate them to continuous additive functionals and intersection local times of general Markov processes. They expect to be able to use them to obtain sample path properties of continuous additive functionals and intersection local times. Important phenomena in our lives, like weather, financial markets, voting patterns and detection of enemy activity are so complex that they can only be modeled as random events called stochastic processes. Nevertheless, although these events are random, they all have certain structures, usually different, that enables us to make good predictions about how they behave, so that we can exploit them or defend ourselves against them. Many probabilists study stochastic processes. Our specialization is local times which is a measure of what are the outcomes of these processes and which outcomes are more or less likely. Our motivation is twofold. One is esthetic, because the underlying mathematics is very beautiful. The other is practical, to provide tools for engineers and scientists engaged in protecting us from devastating weather, controlling destructive market fluctuations, analyzing voter patterns, protecting us from enemy missles...the list of potential applications is endless. Devices and techniques employing the most advanced mathematics contribute to the best of the new growth industries and will aid in keeping our country competitive