Loewner Equation driven by complex-valued functions

Consider the Loewner equation associated to the upper-half plane. Normally this equation is driven by a real-valued function. In this paper, we show that when the driving function is complex-valued with small $1/2$-H\"older norm, the corresponding hull is a quasi-arc, hence is a simple curve. We also study how the hull changes with respect to complex parameters and make a connection between Loewner equation and complex dynamics.