Collaborative Research: Research in Stochastic Processes

点击翻译按钮获取中文摘要

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