Loewner evolution driven by complex Brownian motion

Abstract

We study the Loewner evolution whose driving function is ${\mathit{W}}_{\mathit{t}}={\mathit{B}}_{\mathit{t}}^1\mathbf{+}\mathit{i}{\mathit{B}}_{\mathit{t}}^2$, where $({\mathit{B}}^1,{\mathit{B}}^2)$ 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 ${\mathit{B}}^1$ nor ${\mathit{B}}^2$ 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.