Professors Marcus and Rosen will continue their study of continuity properties of functionals of symmetric Markov processes and their limiting behavior. Using an isomorphism theorem of Dynkin and extensions that they have obtained they will exploit the interplay between high order Wick power and Wick product Gaussian chaos processes and the renormalized self-intersection local time of a large class of Levy processes, to study the phenomenon of ``near-intersections'' of a Levy process and path properties of local times of Levy processes on lines and curves. The processes considered include Brownian motion and stable processes in dimensions one two and three, as well as processes in their domains of attraction. Marcus and Rosen plan to characterize the self-intersection local time functional as the process of zero quadratic variation in the decomposition of an associated Dirichlet process. Given that the self-intersection local time functional has zero quadratic variation it is natural to ask what is its precise p-variation. They plan to investigate this question as well. They expect to be able to use techniques developed in an earlier paper based on the theory of random Fourier series to expand the definition of the self-intersection local time functional by replacing measures by distributions. They will continue their study of continuity properties of high order Gaussian chaos processes and stable moving averages, and level crossings of absolutely continuous infinitely divisible processes. They will also study the multifractal spectrum of occupation measures and intersections of random walks on a four dimensional lattice.
This research deals with fundamental properties of stochastic processes. It exposes a deep relationship between two classes of stochastic processes, Markov processes and Gaussian chaos processes that previously were considered to be unrelated. This relationship leads to increased understanding of the two component processes, an understanding that cannot be obtained by considering the processes separately. These results may have long range applications in the transmission, reception and analysis of data, in quantum physics and in the analysis of financial derivatives.
