Boundaries of escaping Fatou components

Let $f$ be a transcendental entire function and $U$ be a Fatou component of $f$. We show that if $U$ is an escaping wandering domain of $f$, then most boundary points of $U$ (in the sense of harmonic measure) are also escaping. In the other direction we show that if enough boundary points of $U$ are escaping, then $U$ is an escaping Fatou component. Some applications of these results are given; for example, if $I(f)$ is the escaping set of $f$, then $I(f)\cup\{\infty\}$ is connected.