Effect of Random Time Changes on Loewner Hulls

TL;DR

This paper investigates how random time changes affect the geometric properties of Loewner hulls, revealing that such modifications can lead to non-simple curves and providing criteria to determine hull simplicity.

Contribution

It introduces criteria to assess hull simplicity under random time changes and analyzes the impact of inverse stable subordinators on Loewner hulls.

Findings

Time-changed Brownian motions often produce non-simple curves.

Criteria are developed to determine hull simplicity or non-simplicity.

Deterministic results show fast-moving driving functions generate tangential hulls.

Abstract

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.