Potential Theory of Symmetric Stable Processes, p-Laplacian on Trees and Quasiregular Maps

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

ABSTRACT The goal of the research is to study potential theoretic problems arising in the following areas. (1) Symmetric stable processes are discontinuous and nonlocal versions of Brownian motions. The jumps of the processes impose many technical complications, at the same time making the roughness of the boundary invisible; consequently they produce many unexpected properties and interesting questions. Harmonic measures, Martin boundaries and a certain version of boundary Harnack principle will be investigated. (2) On trees of nonregular branching, or on a random Galton-Watson tree, problems on sizes of Fatou sets of bounded p-harmonic functions will be studied. These are discrete analoge of the usual p-harmonic functions. (3) For a quasiregular mapping on the unit ball, the relation between the volume growth of the mapping and size of the set on the unit sphere where the function has asymptotic values will be analyzed. This constitutes a first step towards a very difficult problem of Fatou sets for bounded quasiregular mappings. Brownian motion has been studied at least since Norbert Wiener and has played a central role in probability and potential theory, which in turn are very important in the study of differential equations, heat conductivity, electrostatic potential and fluid dynamics. Recently, there are many problems in physics and mathematical finance and risk estimation which have been modelled and studied successfully with the use of symmetric stable processes -- discontinuous counterpart of the Brownian motions. A symmetric stable process has discontinuous sample paths and heavy tails, while Brownian motion has continuous sample paths and exponentially decaying tails. Therefore many techniques from Brownian motions can not be routinely adapted to these processes. Wide range applications and challenging behaviors of the processes are the motivation behind the proposed research. Wu hopes that knowledge derived from a theoretical study of these processes from an analytical point of view, will give new tools for applications in physics and finance.