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

摘要本研究的目的是研究以下领域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版本。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变得不可见；因此，它们产生了许多意想不到的性质和有趣的问题。本文将研究谐波测度、马丁边界和某种形式的边界哈纳克原理。(2)在非规则分支树或随机高尔顿-沃森树上，研究有界p调和函数的法集大小问题。这些是通常的p-谐函数的离散类比。(3)对于单位球上的拟正则映射，分析了映射的体积增长与函数有渐近值的单位球上集合大小的关系。这是解决有界拟正则映射的Fatousets问题的第一步。布朗运动至少从诺伯特·维纳（Norbert Wiener）开始被研究，并在概率论和势论中发挥了核心作用，而这两种理论在微分方程、导热性、静电势和流体动力学的研究中也非常重要。近年来，利用布朗运动的不连续对应物对称稳定过程成功地对物理和数学金融和风险估计中的许多问题进行了建模和研究。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，许多来自布朗运动的技术不能常规地适应这些过程。该过程的广泛应用和具有挑战性的行为是拟议研究背后的动机。Wu希望，从分析的角度对这些过程进行理论研究所得的知识，将为物理学和金融学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.