Analytic and Geometric Properties of Random Fields

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This research is concerned with the development of a systematic approach to the study of analytic and geometric properties of random fields. Special emphasis is placed on Gaussian and stable random fields such as the Brownian sheet, stable sheets and additive Levy processes. The investigators wish to continue their study of precise quantitative connections between the aforementioned random fields and the theory of capacities, as well as potential theory for general Markov type random sets. They believe that these connections will yield detailed analytic and geometric information about the random fields in question. Amongst other things, two long-standing open problems are emphasized: one on the potential theory of the Brownian sheet, and the other in the theory of random coverings. The investigators also plan to develop and advance canonical techniques for fractal and multifractal analysis of a large class of multiparameter Levy processes, as well as the Brownian sheet. They expect these techniques will, in turn, be useful in studying complex random fields and processes.The Gaussian and stable random fields considered in this project play a prominent role in many areas of pure and applied mathematics, statistics, mathematical physics, medical imaging, ecology, geology, geophysics, oceanography, hydrology, as well as mathematical finance. This research is concerned with developing and introducing various analytic and geometric tools that will lead to a better understanding of geometric problems for random fields, as well as help promote their future applicability.