Mathematical Sciences: Stochastic Processes

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9503519 Marcus Abstract The investigators will continue their study of continuity properties of additive functionals of Markov processes and their limiting behavior. Using an isomorphism theorem of Dynkin and results from the theory of probability in Banach spaces and random Fourier series they will explore relationships between continuous additive functionals of Markov processes and Gaussian processes and chaoses and random Fourier series. In this way they expect to be able to use known results about each of the different processes to obtain new results about the other ones as they did in their work on local times and Gaussian processes. They also plan to use techniques developed in their earlier work to study the number of intersections of random walks, particularly in the critical and super-critical cases. Motivated by the importance of a special class of second order Gaussian chaoses related tm Wick squares in the study of positive continuous additive functionals, they will try to find necessary and sufficient conditions for the continuity of these chaoses. They will also attempt to use self-collision local times of superprocesses to construct a self-interacting measure valued diffusion and study the large deviations of additive functionals of super Brownian motion. This research deals with fundamental properties of stochastic processes and has potential application in all areas that deal with random phenomena. The investigators will consider data that evolves in time in a random fashion, such as electrical signals received from distant satellites or dollar values in financial transactions. In particular they will consider the amount of time the data takes specific values or is in a small, heavily weighted, range of values and how this amount of time varies as the set of specific values vary. They plan to do this for a wide class of processes called Levy processes. These processes have often been thought of as models for many important practical applications but, so far, they have b een considered too difficult to analyze very deeply. The authors plan to use several new and powerful methods, recently developed by themselves and others, to accomplish this deep analysis.