Deformation of 2-dimensional diffusion processes which preserves recurrence

黎曼曲面的均匀化定理认为，非紧致单连通曲面上的复杂结构等价于复平面上的标准复杂结构或单位圆盘上的标准复杂结构。在布朗运动方面，这种分类完全是由过程在域上的递归来描述的。在复平面上，对于不等同于标准结构的复结构，其beltrami -系数的L^∞范数必须为1，也就是说，如果对应的布朗运动是暂态的，则其beltrami -系数的L^∞范数为1。然而，这并不意味着复杂平原上每一个贝尔特拉米系数具有单位L^∞范数的复杂结构都对应于瞬态布朗运动。本研究项目的目的就是研究这种复杂的结构。可以构造具有临界点的单参数复杂结构族，在临界点处递归性和暂态性相互转换。本文所讨论的复杂结构模型是用闭合单元盘来参数化的，参数的模描述了相应beltrami系数的L^∞范数。当参数位于开式单元盘时，规定的复杂结构与标准结构等效。在单位圆盘的边界上，除了一个点外，复合结构与单位圆盘上的标准结构等效，因此布朗运动是瞬态的。在临界点处，复杂结构是循环的，但保持循环的变形自由度相对较低。进一步研究的一个自然方向是：研究模群对beltrami -系数空间的作用和模群对边界的作用。

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.