Loewner evolution driven by complex Brownian motion (with simulations by Minjae Park)

TL;DR

This paper investigates a generalized Loewner evolution driven by complex Brownian motion, revealing distinct behaviors from classical SLE, including phase transitions characterized by the Brownian motions' covariance and explicit phase boundary formulas.

Contribution

It introduces a new complex Brownian motion-driven Loewner evolution model and analyzes its phase behavior, extending the understanding of SLE-like processes with complex parameters.

Findings

Different behavior from ordinary SLE, with interior points in the hulls.

Identification of three distinct phases: measure-zero, swallowed points, space-filling.

Explicit phase boundary formulas, especially for zero correlation case.

Abstract

We study the Loewner evolution whose driving function is $W_t = B_t^1 + i B_t^2$, where $(B^1,B^2)$ 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 $B^1$ nor $B^2$ is identically equal to zero, then the set of points disconnected from $\infty$ by the Loewner hull has non-empty 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.