November 2023 Loewner evolution driven by complex Brownian motion
Ewain Gwynne, Joshua Pfeffer, Minjae Park
Author Affiliations +
Ann. Probab. 51(6): 2086-2130 (November 2023). DOI: 10.1214/23-AOP1639


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.

Funding Statement

E.G. was partially supported by a Clay research fellowship.
M.P was partially supported by NSF Award DMS-1712862.
J.P. was partially supported by a National Science Foundation Postdoctoral Research Fellowship under Grant No. 2002159.


We thank an anonymous referee for helpful comments on an earlier version of this article. We thank Tom Kennedy and Scott Sheffield for helpful discussions. We thank Steffen Rohde for sharing with us an old email and Mathematica file from his work on this topic with Schramm in 2006.


Download Citation

Ewain Gwynne. Joshua Pfeffer. Minjae Park. "Loewner evolution driven by complex Brownian motion." Ann. Probab. 51 (6) 2086 - 2130, November 2023.


Received: 1 April 2022; Revised: 1 June 2023; Published: November 2023
First available in Project Euclid: 12 November 2023

Digital Object Identifier: 10.1214/23-AOP1639

Primary: 30C20 , 60J67

Keywords: Brownian motion , complex driving function , conformal mapping , Loewner evolution , Schramm–Loewner evolution

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 6 • November 2023
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