Investigating Hulls From The Loewner Equation

The Loewner equation gives a correspondence between real functions and sets in the upperhalf-plane called Loewner hulls. This dissertation has three major parts.First, Kager, Nienhuis, and Kadanoff conjectured that the hull generated from the Loewner equation driven by two constant functions with constant weights could be generated by a single rapidly and randomly oscillating function. We prove their conjecture and generalize to multiple continuous driving functions. In the process, we generalize to multiple hulls a result of Roth and Schleissinger that says multiple slits can be generated by constant weight functions. The proof gives a simulation method for hulls generated by the multiple Loewner equation.Second, 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.Third, Lind proved that if the Lipschitz one-half norm of the driving function is less than 4, then the generated curve is simple. After this, Lind and Rohde showed that if a Lipshitz one-half driving function generates a spacefilling curve then the norm is greater than 4.0001. This bound is not optimal. We discuss our work towards finding the optimal lower bound on the Lipschitz one-half norm of driving functions that generate a spacefilling curve