Research in Stochastic Processes

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Professors Marcus and Rosen have been studying the relationship between the local times of strongly symmetric Markov processes and Gaussian processes for many years. They have obtained many interesting results about local times which they have published in more than a dozen papers and a monograph. Until recently their work was based on an isomorphism theorem of Dynkin, which is difficult to prove and to apply. In the last two years this has all changed. Together with Professors Eisenbaum, Kaspi and Shi, they have obtained new, simple isomorphisms relating local times and Gaussian processes and have greatly simplified and clarified their early work. They have also obtained many new results; the most significant is a simplified version of Ray's theorem on the local times of diffusions. They will continue this work to generalize the scope of Ray's theorem to consider local times of processes which are not continuous. They will apply their new results and techniques to consider other properties of Markov processes that can be studied through their local times. They also plan to extend their results to more general classes of continuous additive functionals of strongly symmetric Markov processes by comparing them to Gaussian chaos processes. Professor Marcus will continue his studies of sample path properties of infinitely divisible moving average processes. These process are fundamental in applied mathematics. They appear to have remarkable smoothness properties and to behave better than similarly defined Gaussian processes. This surprising observation will be investigated. Professor Rosen plans to study the time needed for a simple random walk to visit each point on a finite graph. The case of the two dimensional lattice torus is particularly challenging. He intends to study this discrete problem by relating it to a continuous one concerning Brownian motion on the two dimensional torus. This in turn will lead to the analysis of `late points', those points whose approach by the Brownian path takes an unusually large amount of time. Professor Rosen also plans to study points of infinite multiplicity on the path of planar Brownian motion. This research deals with fundamental properties of stochastic processes and has potential applications in all areas that deal with random phenomena. Generally speaking phenomena that evolve in time do so in a random fashion. Examples are the Dow Jones average, data on global warming or communication with satellites. Of particular importance is the amount of time that a process takes a specific value. This is studied in terms of the local time of the process. In this proposal the local times of Markov processes will be investigated by means of associated Gaussian processes. Until very recently these two important classes of stochastic processes, Markov processes and Gaussian processes, were considered to be essentially unrelated. Professors Marcus and Rosen have shown that they are intimately related and are searching for a unified theory for these important processes.