Revista Matemática Iberoamericana


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Published online first: 2019-10-15
DOI: 10.4171/rmi/1148

Effect of random time changes on Loewner hulls

Kei Kobayashi[1], Joan Lind[2] and Andrew Starnes[3]

(1) Fordham University, New York, USA
(2) The University of Tennessee, Knoxville, USA
(3) University of Hartford, West Hartford, USA

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.

Keywords: Loewner evolution, random time change, inverse subordinator, time-changed Brownian motion

Kobayashi Kei, Lind Joan, Starnes Andrew: Effect of random time changes on Loewner hulls. Rev. Mat. Iberoam. Electronically published on October 15, 2019. doi: 10.4171/rmi/1148 (to appear in print)