Mathematical Sciences: Research on Representing Measures and Boundary Behavior in Potential Theory

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Author: loeb@math.uni-frankfurt.de at NOTE Date: 3/5/96 4:43 AM Priority: Normal TO: jjenkins at nsf11 Subject: ------------------------------- Message Contents ------------------------------- This e-mail letter is sent by Peter A. Loeb from Frankfurt, Germany where he is a visitor for the month of March. Abstract of Peter A. Loeb's project The proposed research will continue Professor Peter Loeb's work on representing measures and ideal boundaries in potential theory. In particular, Loeb plans to continue research on a Martin-like boundary which is regular almost everywhere with respect to harmonic measure. An important tool in this research is the technique developed with J. Bliedtner that reduces the production of Radon-Nikodym derivatives as limits to a special case. This reduction casts new light on the martingale convergence theorem and significantly simplifies measure differentiation theorems as well as the treatment of fine limit theorems in probability and potential theory. A major consequence is the existence of simple boundary approach neighborhood systems which, after fixing a suitable normalization, are the "best" possible in terms of producing Radon-Nikodym derivatives as limits at the boundary. The existence of such "best" systems had not been known even for harmonic functions on the unit disk. Indeed, previous results in the literature suggested that no such system could exist. In related, ongoing research, Loeb has improved the celebrated Besicovitch and Morse covering theorems from geometric measure theory by strengthening the results and simplifying the proofs, including a demonstration that for any norm, the best constant (in terms of all known proofs) for the Besicovitch result is a packing constant. In all of his research, Professor Loeb will continue to employ nonstandard models. These are mathematical structures with infinitely large and infinitely small numbers. Even when the use of such structures is removed from the final proofs of results, they are extremely helpful in the discovery process. Loeb has recently used nonstandard models to optimize the coverings of Besicovitch's theorem, and with Bliedtner he is now using these methods to extend the notion of radial limits to general potential theoretic settings. The results of this research that have been obtained to date, and the results that will be established in the proposed research all lie at the heart of mathematical analysis and probability theory. These are fundamental areas of mathematics used by scientists and engineers to form the conceptual models of the phenomena with which they work. The thought process itself can only be couched in terms of mathematics, and the mathematics being used in turn yields new insight into the phenomena being studied. Therefore, the simplification and extension of the basic mathematical tools achieved by the proposed research will resonate in all of the areas of science and engineering that use these tools. Of particular importance for future developments will be the successful employment of mathematical structures with infinitely large and infinitely small numbers. These structures have a great deal to offer in all areas of science and engineering because of the simplicity and power inherent in their use. An important construction associated with these structures are measure spaces, now called "Loeb spaces" in the literature, which allow the treatment of infinite probability phenomena using methods appropriate for finite sets. Researchers including Loeb have obtained new results with these methods in areas such as the study of partial differential equations, probability theory, research in economics, and the study in physics of laws governing the motion of gases.