Mathematical Sciences: Stochastic Processes

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Professors Marcus and Rosen will continue their investigations of the relationship between Gaussian processes and the local times of symmetric Markov processes put into evidence by the Dynkin Isomorphism Theorem. A compact form of this Theorem will be used in an attempt to obtain Borell type inequalities for local times. Analogous to their recent work which shows the equivalence of sample path properties of the local times of a symmetric Markov process and an associated Gaussian process they will study the relationship between continuous additive functionals of a symmetric Markov process indexed by measures and an associated class of Gaussian chaoses indexed by measures. This will require an investigation into measure indexed Gaussian processes and chaoses in general. They will also try to extend their recent work on the law of the iterated logarithm for the local times of symmetric Levy processes and random walks to cover the interesting case of recurrent random walks on a two dimensional integer lattice. Professor Marcus and Rosen will continue their investigations of the relationship between Gaussian Processes and the local times of symmetric Markov processes. Until recently these stochastic processes were not considered to be related at all, except in the case of Brownian motion, a relationship was implied by a recent isomorphism theorem of Dynkin. This isomorphism has been exploited by Professors Marcus and Rosen to show that there are deep connections between these two fundamental classes of stochastic processes, both of which serve as models in countless applications to continuous time random phenomena. This research could lead to significant new knowledge about our random environment.