Collaborative Research: Research in Stochastic Processes

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Professors Marcus and Rosen will continue their research on the relationship between Gaussian processes and the local times of related strongly symmetric Markov processes. This is the subject of their Cambridge University Press book, Markov Processes, Gaussian Processes and Local Times, which was published in October 2006. In writing this book they solved many problems and uncovered many new ones. They are particularly interested in finding a heuristic explanation, based on sample path properties, of the fact that Gaussian processes with infinitely divisible squares are precisely those Gaussian processes with covariance that is the zero potential density of a strongly symmetric Markov process. They will also continue their program of using Gaussian process techniques to discover new sample path properties of local times, such as central limit theorems for the moduli of continuity of local times of Markov processes. In continuing to explore the interplay between Gaussian and Markov processes they will also consider non-normal central limit theorems for the moduli of continuity of Gaussian processes with convex covariance functions as a first step towards obtaining similar properties for local times. Many important phenomena, such as weather patterns or the behavior of the stock market, are so complex that the only way to study them is to consider them as random, or stochastic, processes. Mathematical models of stochastic processes are studied to give insight into the physical phenomena that they represent. On subtle property of a stochastic process is the amount of time realizations of the process spend at the possible values that it can take. This property is called the local time of the process. This proposal is to continue research on the local times of symmetric Markov processes.