Research in Stochastic Processes

Cuny City College, New York NY

Investigators

Abstract

0404952 Marcus 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.