Loewner hulls are determined by their real-valued driving functions. We study the geometric effect on the Loewner hulls when the driving function is composed with a random time change, such as the inverse of an $\alpha$-stable subordinator. In contrast to SLE, we show that for a large class of random time changes, the time-changed Brownian motion process does not generate a simple curve. Further we develop criteria which can be applied in many situations to determine whether the Loewner hull generated by a time-changed driving function is simple or non-simple. To aid our analysis of an example with a time-changed deterministic driving function, we prove a deterministic result that a driving function that moves faster than $at^r$ for $r \in (0, 1/2)$ generates a hull that leaves the real line tangentially.

中文翻译：

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随机时间变化对Loewner船体的影响

洛纳船体由其实值驱动功能确定。当驱动函数由随机时间变化（例如，$ \ alpha $稳定的从属子的逆函数）组成时，我们研究了对Loewner船体的几何影响。与SLE相比，我们表明，对于一大类随机时间变化，时变布朗运动过程不会生成简单曲线。此外，我们开发了可在许多情况下应用的标准，以确定由时变驱动函数生成的Loewner船体是简单的还是非简单的。为了帮助我们分析随时间变化的确定性驱动函数的示例，我们证明了确定性结果，即对于$ r \ in（0，1/2）$，运动速度快于$ at ^ r $的驱动函数会生成一个船体切线地留下实线。