带形区域上的通弦SLE和反向SLE的一些性质

发布时间：2018-05-23 17:11

本文选题：SLE_κ + 壳　；参考：《广西民族大学》2016年硕士论文

【摘要】：随机Loewner演变(简称SLE或SLE_κ)是通过解驱动函数为时间改变的布朗运动的Loewner微分方程来描述的一类带有一个参数κ的共形不变随机分形曲线族.对SLE_κ的研究,从上半平面和单位圆盘上的典型SLE_κ被推广到一般区域的情形.本文的主要工作如下:第一,讨论了带形区域上SLE_κ壳的性质.利用带形区域上SLE_κ的性质与Schwarz反射原理,给出了R-对称共形映射与带形区域内壳的关系;得到了带形区域内由一对不相交的壳组成集合与Loewner共形映射之间的关系;导出了R-对称共形映射的提升在带形区域的壳空间内以及带形区域的壳对空间上的相关映射是连续的.这就将上半平面上SLE壳的有关性质推广到了带形区域的情形.第二,研究了带形区域上反向SLE_κ的性质.应用Leowner方程定义带形区域上反向SLE_κ、向前Loewner链和反向Loewner链;讨论了带形区域上向前Loewner链和反向Loewner链的性质,得到了向前Loewner壳与向前Loewner链之间以及反向Loewner壳和反向Loewner链之间的关系;讨论了涉及一些简单曲线的随机共形粘合.
[Abstract]:Stochastic Loewner evolution (SLE or SLEK) is a class of conformal invariant random fractal curves with a parameter 魏, which is described by solving the Loewner differential equation of Brownian motion with time-varying driving function. The study of SLEK is extended from the typical SLE- 魏 on the upper half plane and the unit disk to the general domain. The main work of this paper is as follows: first, we discuss the properties of SLE- 魏 shells in the zonal region. By using the properties of SLE- 魏 and the Schwarz reflection principle, the relation between the R- symmetric conformal mapping and the inner shell of the band region is given, and the relation between the Loewner conformal mapping and the set of disjoint shells in the zonal region is obtained. It is derived that the lifting of R- symmetric conformal mapping is continuous in the shell space of the band region and the correlation mapping between the shell and the space of the belt region. This extends the properties of the SLE shell on the upper half plane to the case of a zonal region. Secondly, we study the properties of reverse SLE- 魏 in the zonal region. The Leowner equation is used to define the reverse SLE- 魏, forward Loewner chain and reverse Loewner chain in the band region, and the properties of the forward Loewner chain and reverse Loewner chain in the band region are discussed, and the relations between the forward Loewner shell and the forward Loewner chain, as well as the reverse Loewner shell and the reverse Loewner chain are obtained. Random conformal bonding involving some simple curves is discussed.
【学位授予单位】：广西民族大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O211.6

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