论文标题
由复杂的布朗运动驱动的Loewner进化（Minjae Park进行了模拟）
Loewner evolution driven by complex Brownian motion (with simulations by Minjae Park)
论文作者
论文摘要
我们研究Loewner Evolution的驾驶功能为$ W_T = B_T^1 + I B_T^2 $，其中$（B^1，B^2）$是一对带有给定协方差矩阵的Brownian Motions。可以将此模型视为具有复杂参数值的Schramm-loewner进化（SLE）的概括。我们表明，我们的Loewner演变与普通SLE的行为有很大不同。例如，如果$ b^1 $和$ b^2 $都不等于零，则Loewner Hull与$ \ infty $断开的积分的集合每次都有非空的内部装饰。我们还表明，我们的模型表现出类似于SLE相似的三个阶段：船体的Lebesgue测量为零的相位，一个相位，该阶段吞咽但未被船体击中，而船体在空间填充空间的相位。相边界是根据显式积分的符号表示的。当两个布朗运动的相关性为零时，这些边界具有简单的封闭形式。
We study the Loewner evolution whose driving function is $W_t = B_t^1 + i B_t^2$, where $(B^1,B^2)$ is a pair of Brownian motions with a given covariance matrix. This model can be thought of as a generalization of Schramm-Loewner evolution (SLE) with complex parameter values. We show that our Loewner evolutions behave very differently from ordinary SLE. For example, if neither $B^1$ nor $B^2$ is identically equal to zero, then the set of points disconnected from $\infty$ by the Loewner hull has non-empty interior at each time. We also show that our model exhibits three phases analogous to the phases of SLE: a phase where the hulls have zero Lebesgue measure, a phase where points are swallowed but not hit by the hulls, and a phase where the hulls are space-filling. The phase boundaries are expressed in terms of the signs of explicit integrals. These boundaries have a simple closed form when the correlation of the two Brownian motions is zero.
