This week we study the connections between the material we?ve covered and harmonic measure. The heuristic definition (though we will not rely on it) is, given a domain, the harmonic measure of a subset of its boundary is the probability that a Brownian motion starting from a fixed point first leaves the domain through this set. It turns out that there is a strong connection between geometry of the domain and the behavior of its harmonic measure. We will focus on harmonic measure in non-tangentially accessible domains, and will rely on results from nearly every previous week of lectures.
- Notes (alternatively, read Section 4 of JK82, KPT09). *Note: the notes are quite long this week, so the videos might actually be more concise.
- Videos: Intro to Harmonic Measure, Harmonic Functions, Green’s Function, NTA Domains, Rectifiability of Harmonic Measure, The 2-Phase Problem
Optional: Overview of Results on Rectifiability of Harmonic Measure
- Exercises (Prepare any one of these.)