Anne Schreuder: Loewner Chains and their driving measures

Abstract:

Since Charles Loewner’s seminal paper in 1923 Loewner chains have been a powerful tool in Mathematics. Originally, applied to distortion estimates for univalent functions such as Bieberbach’s conjecture/de Branges theorem, Loewner chains have been rediscovered as a method to describe random curves, e.g. Schramm-Loewner Evolutions, and random growth models such as Diffusion Limited Aggregation (DLA) or Hasting-Levitov type growth models. This works as by the Riemann Mapping Theorem there is a bijection between Loewner chains and (continuously) growing simply connected sets in the plane.

Crucially, there is also a bijection between Loewner chains and finite Borel measures which constitutes an analytic representation theorem. One direction of this bijection is given by the famous Loewner-Kufarev Equation. We were interested in the converse question: How can the driving measure be obtained from its Loewner chain. In particular, we obtain explicit formulations of the driving measure and an expression for the density of the driving measure.