Harmonic measure

Summary

The harmonic measure of a domain, seen from a point $z$, is a probability measure on the boundary of the domain. It is the measure that represents the value at $z$ of the solution of the Dirichlet problem in terms of the boundary data. Equivalently, it is the distribution of the first exit point of a Brownian motion started at $z$. It is invariant under conformal maps, and it is a central tool for estimating harmonic and holomorphic functions near the boundary.