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

本研究的目的是研究在以下方面可能出现的理论问题。(1)对称稳定过程是布朗运动的不连续和非局部形式。过程的跳跃带来了许多技术复杂性，同时使边界的粗糙程度不可见；因此，它们产生了许多意想不到的性质和有趣的问题。调和度量、马丁边界和某种形式的边界哈纳克原理将被研究。(2)在非正则分枝树或随机Galton-Watson树上，研究有界p-调和函数的Fatou集的大小问题。这些都是通常的p-调和函数的离散模拟。(3)对于单位球上的拟正则映射，分析了映射的体积增长与单位球上函数具有渐近值的集合的大小之间的关系。这是向有界拟正则映射的Fatouset问题迈出的第一步。布朗运动至少从Norbert Wiener开始就被研究过，并在概率论和位势理论中发挥了中心作用，而这些理论在微分方程、导热、静电势和流体动力学的研究中又是非常重要的。近年来，在物理、数学、金融和风险估计等领域有许多问题已经被成功地用对称稳定过程--布朗运动的不连续对应过程--来模拟和研究。对称稳定过程具有不连续的样本路径和重尾，而布朗运动具有连续的样本路径和指数衰减的尾巴。因此，布朗运动的许多技术不能常规地适应这些过程。这些过程的广泛应用和具有挑战性的行为是本研究背后的动机。吴希望，从分析的角度对这些过程进行理论研究所获得的知识，将为物理和金融领域的应用提供新的工具。

ABSTRACTThe goal of the research is to study potential theoretic problemsarising in the following areas. (1) Symmetric stable processes arediscontinuous and nonlocal versions of Brownian motions. The jumps ofthe processes impose many technical complications, at the same timemaking the roughness of the boundary invisible; consequently theyproduce many unexpected properties and interesting questions.Harmonic measures, Martin boundaries and a certain version of boundaryHarnack principle will be investigated. (2) On trees of nonregularbranching, or on a random Galton-Watson tree, problems on sizes ofFatou sets of bounded p-harmonic functions will be studied. Theseare discrete analoge of the usual p-harmonic functions. (3) Fora quasiregular mapping on the unit ball, the relation between thevolume growth of the mapping and size of the set on the unit spherewhere the function has asymptotic values will be analyzed. Thisconstitutes a first step towards a very difficult problem of Fatousets for bounded quasiregular mappings.Brownian motion has been studied at least since Norbert Wiener and hasplayed a central role in probability and potential theory, which inturn are very important in the study of differential equations, heatconductivity, electrostatic potential and fluid dynamics. Recently,there are many problems in physics and mathematical finance and riskestimation which have been modelled and studied successfully with theuse of symmetric stable processes -- discontinuous counterpart of theBrownian motions. A symmetric stable process has discontinuoussample paths and heavy tails, while Brownian motion has continuoussample paths and exponentially decaying tails. Therefore manytechniques from Brownian motions can not be routinely adapted to theseprocesses. Wide range applications and challenging behaviors of theprocesses are the motivation behind the proposed research. Wu hopesthat knowledge derived from a theoretical study of these processesfrom an analytical point of view, will give new tools for applicationsin physics and finance.