Loewner Evolution Driven by Complex Brownian Motion
The Analysis and Geometry of Random Spaces March 28, 2022  April 01, 2022
Location: SLMath: Eisenbud Auditorium, Online/Virtual
Loewner Evolution Driven By Complex Brownian Motion
We consider 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 SchrammLoewner 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 complements of the Loewner hulls are not connected. 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 spacefilling. 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. Based on joint work with Josh Pfeffer, with simulations by Minjae Park.
Loewner Evolution Driven by Complex Brownian Motion

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Loewner Evolution Driven By Complex Brownian Motion
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