Planar Brownian motion and complex analysis

I intend to study in depth a number of related problems concerning both Brownian motion and complex analysis. These include questions about Brownian exit time, conformally invariant processes such as Stochastic Loewner Evolution, and the fundamentals of complex analysis. Many of these questions are at the forefront of modern probability theory, given the recent explosion of interest in conformal invariance of processes. It is anticipated that a number of high quality publications will result, bringing the questions considered into a position of prominence in the fields of probability and analysis, and bringing international attention to Australia as a hub of important research.