11 - Harmonic Measure

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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.