Quasiconformal maps and harmonic measure

Harmonic measure (of a boundary of a domain) is a fundamental notion in complex analysis, and can be defined in many ways: as an elctrostatic equilibrium distribution, as the exit probability for the Brownian motion, as a measure providing a solution to the Dirichlet boundary value problem. For simply connected planar domains harmonic measure is the image of length under the uniformization map from the unit disk, ans so many important questions in complex analysis can be reduced to the investigations into its multifractal properties, i.e. the study of the sets, where the measure staisfies a prescribed power law. We will discuss possible approaches to these questions using quasiconformal maps and holomorphic motions. Motivation comes from dynamical systems and mathematical physics, but as a result we return again to questions from classical complex analysis.
We will tell about our project in progress with Kari Astala and Istvan Prause, which aims at describing the dimensional structure of the harmonic measure in the plane.