Deformation of 2-dimensional diffusion processes which preserves recurrence

Riemann曲面的单值化定理证明了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。在布朗运动的分类是完全描述的递归过程的域。对于那些与标准结构不等价的复杂结构，在复平面上贝尔特拉米系数的L^∞-模必须为1，换句话说，如果相应的布朗运动是瞬时的，则贝尔特拉米系数的L^∞-模为1。然而，这并不意味着复平面上Beltrami系数具有单位L^∞范数的每个复结构都对应于瞬态布朗运动。本研究项目的目的是研究这种复杂的结构。人们可以构造一个具有临界点的单参数复杂结构族，在临界点处，递归性和瞬态性切换到另一个。本文所讨论的复杂结构模型是用封闭的单位圆盘参数化的，参数的模描述了相应Beltrami系数的L^∞-范数。当参数位于开单位圆盘内时，指定的复结构与标准复结构等价。在单位圆盘的边界上，除一点外，复结构与单位圆盘上的标准结构等价，因此布朗运动是瞬态的。在临界点的复杂结构是经常性的，但保持复发的变形的自由度相对较低。进一步研究的一个自然方向是：研究模群在Beltrami系数空间上的作用和模群在边界上的作用。

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.