Effect of Random Time Changes on Loewner Hulls

2018/02/26 by Kobayashi, Kei, Lind, Joan, Starnes, Andrew
#Complex Variables (math.CV) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1802.09466

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 α-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 ∈ (0,1/2) generates a hull that leaves the real line tangentially.

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