Cornell Math - MATH 712, Spring 2006
MATH 712: Planar Harmonic Measure (Spring 2006)
Instructor: Luke Rogers
The goal of this course is to introduce planar harmonic measure and its connections to potential theory, conformal mappings and Brownian motion, and then to study some results relating harmonic measure to other measures on the boundary of a domain, particularly to length and other Hausdorff measures.
The simplest example of harmonic measure may be seen in the context of the classical Dirichlet problem on the unit disc. Given appropriate boundary data (for example a continuous function f) we may find an extension of f which is harmonic on the interior of the disc by integration against the Poisson kernel. This may be interpreted as showing there is a measure w(z) on the boundary (this is the harmonic measure with respect to z) such that the harmonic extension at z is obtained by integrating f with respect to this measure. This result may be transferred to all simply-connected Jordan domains via the Riemann mapping theorem, and then with some limitations, to more general domains. For the case of infinitely-connected domains, we will see two proofs of this, the first of which uses exhaustion by simple domains and weak* convergence of the harmonic measures, and the second of which uses Brownian motion. The latter approach is a particularly good source of intuition for harmonic measure, as it identifies the harmonic measure as the hitting probability of a Brownian trajectory started at z.
Certain aspects of the geometry of the boundary of a domain are reflected in the behavior of the harmonic measure for points z near the boundary. The simplest example of this may be seen by examining the effect of the Riemann mapping of the disc to a sector, in which length measure near the vertex is distorted by a power. For this reason, it is interesting to know information about the support of the harmonic measure and the size of the sets on which the harmonic measure has particular scaling behaviors. I intend to give a proof of Makarov's Law of the Iterated Logarithm (for Bloch functions) which gives an explicit gauge for the support of the harmonic measure, to survey some results on a conjecture due (in various forms) to Brennan-Carleson-Jones-Kraetzer-Binder about the scaling behaviors of conformal maps, and (if time permits) to examine some square-function methods of Bishop-Jones which express the regularity of sets in terms of Schwartzian derivatives.