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

本研究的目的是研究以下几个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上的复杂性，同时使边界的粗糙度不可见;因此它们产生了许多意想不到的性质和有趣的问题。调和测度，Martin边界和边界Harnack原理的某个版本将被研究。 (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。这些是通常p-调和函数的离散类比。(3)对于单位球上的拟正则映射，分析了在函数具有渐近值的情况下，映射的体积增长与单位球上集合的大小之间的关系。 布朗运动至少从Norbert Wiener开始就被研究，并在概率论和位势论中起着中心作用，而这又在微分方程、热传导、静电势和流体动力学的研究中起着非常重要的作用。 近年来，在金融学、金融数学和风险估计等领域中，许多问题都已成功地用布朗运动的不连续对称稳定过程进行了建模和研究。 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。 因此，布朗运动的许多技术不能常规地适用于这些过程. 广泛的应用和具有挑战性的行为的过程是背后的动机提出的研究。 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.