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Deformation of 2-dimensional diffusion processes which preserves recurrence

保持重现的二维扩散过程的变形

基本信息

批准号：
12640127
负责人：
IWATA Koichiro
金额：
$ 1.86万
依托单位：
HIROSHIMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

The uniformization theorem foe Riemann surfaces claims that complex structures on non-compact simply connected surfaces are equivalent to either the standard complex structure on the complex plain or the one on the unit disk. In terms of Brownian motions this classification is completely described by the recurrence of the process on the domain. On the complex plain the L^∞-norm of Beltrami-coefficients must be 1 as for those complex structures not equivalent to the standard one, in other words, if the corresponding Brownian motion is transient then the L^∞-norm of Beltrami-coefficient is 1. However this does not mean that every complex structure on the complex plain whose Beltrami-coefficient has unit L^∞-norm corresponds to transient Brownian motions. The aim of the present research project is to study such complex structures. One can construct a one-parameter family of complex structures with critical point and at the critical point the recurrence and the transience switch to the other. The model of the complex structures discussed in the present research is parameterized by the closed unit disk and the modulus of the parameter describes the L^∞-norm of the corresponding Beltrami-coefficient. When the parameter lies in the open unit disk the prescribed complex structure is equivalent to the standard one. On the boundary of the unit disk except for one point the complex structure is equivalent to the standard one on the unit disk and thus the Brownian motion is transient. At the critical point the complex structure is recurrent but the freedom of deformation that preserves recurrence is relatively low. A natural direction for further direction is as follows : Study the action of the modular group on the space of Beltrami-coefficients and the action of the modular group on the boundary.

Riemann曲面的均匀化定理认为，非紧单连通曲面上的复结构等价于复平面上的标准复结构或单位圆盘上的标准复结构。在布朗运动方面，这种分类完全由区域上过程的常返性来描述。在复平面上，对于不等价于标准布朗运动的复杂结构，Beltrami系数的L∞范数必须为1，换言之，如果相应的布朗运动是瞬变的，则Beltrami系数的L∞范数为1。但这并不意味着Beltrami系数具有单位L∞范数的复杂平面上的每个复杂结构都对应于瞬变布朗运动。本研究项目的目的就是研究这种复杂的结构。可以构造一个具有临界点的单参数复杂结构族，并且在临界点上递归和瞬变切换到另一个。本文所讨论的复杂结构模型是用闭单位圆盘来描述的，参数的模用来描述相应Beltrami系数的L∞范数。当参数位于开放的单位圆盘上时，规定的复杂结构等价于标准结构。在单位圆盘的边界上，除一点外，复数结构与单位圆盘上的标准结构等价，因此布朗运动是瞬变的。在临界点，复杂的结构是重复的，但保持重复的变形自由度相对较低。进一步研究模群在Beltrami系数空间上的作用和模群在边界上的作用是进一步发展方向的一个自然方向。