Loewner evolution driven by complex Brownian motion

Abstract

We study the Loewner evolution whose driving function is Wt=Bt1+iBt2, where (B1,B2) is a pair of Brownian motions with a given covariance matrix. This model can be thought of as a generalization of Schramm–Loewner evolution (SLE) with complex parameter values. We show that our Loewner evolutions behave very differently from ordinary SLE. For example, if neither B1 nor B2 is identically equal to zero, then the set of points disconnected from ∞ by the Loewner hull has nonempty interior at each time. We also show that our model exhibits three phases analogous to the phases of SLE: a phase where the hulls have zero Lebesgue measure, a phase where points are swallowed but not hit by the hulls and a phase where the hulls are space-filling. The phase boundaries are expressed in terms of the signs of explicit integrals. These boundaries have a simple closed form when the correlation of the two Brownian motions is zero.