Effect of Random Time Changes on Loewner 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 $a{t}^r$ for $r\in (0,1/2)$ generates a hull that leaves the real line tangentially.

arXiv comment26 pages, 8 figures, keywords and acknowledgments updated

References in corpus (8)

• Fractional Cauchy problems on bounded domains
• Stochastic Calculus for a Time-changed Semimartingale and the Associated Stochastic Differential Equations
• A Sharp Condition for the Loewner Equation to Generate Slits
• SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations
• First passage time processes and subordinated SLE
• Schramm-Loewner Equations Driven by Symmetric Stable Processes
• Stochastic Loewner Evolution Relates Anomalous Diffusion and Anisotropic Percolation
• Singular and tangent slit solutions to the Loewner equation