 Harmonic measure

The harmonic measure of a boundary set $A\subseteq \partial D$ viewed from $x\in D$ is the probability that Brownian motion started at $x$ first exits $D$ through $A$. As a function of $x$, it is harmonic in $D$.

Harmonic measure is a concept in mathematical analysis, particularly in potential theory and complex analysis. It is associated with harmonic functions, which are functions that satisfy Laplace's equation. Here are some key points to understand harmonic measure: 1. **Harmonic Functions**: A function \( u \) is harmonic in a domain if it is twice continuously differentiable and satisfies Laplace's equation, i.e., \( \nabla^2 u = 0 \).