In my talk I will discuss Loewner chains whose driving functions are complex Brownian motions with general covariance matrices. This extends the notion of Schramm-Loewner evolution (SLE) by allowing the driving function to be complex-valued and not just real-valued. We show that these Loewner chains exhibit the same phases (simple, swallowing, and space-filling) as SLE, and we explicitly characterize the values of the covariance matrix corresponding to each phase. In contrast to SLE, we show that the evolving left hulls are a.s. not generated by curves, and that they a.s. disconnect each fixed point in the plane from infinity before absorbing the point.
This talk is based on a joint work with Ewain Gwynne.