  We study the dynamical behaviour of points in the boundaries of simply
connected invariant Baker domains $U$ of meromorphic maps $f$ with a finite
degree on $U$. We prove that if $f|_U$ is of hyperbolic or simply parabolic
type, then almost every point in the boundary of $U$ with respect to harmonic
measure escapes to infinity under iteration. On the contrary, if $f|_U$ is of
doubly parabolic type, then almost every point in the boundary of $U$ with
respect to harmonic measure has dense forward trajectory in the boundary of
$U$, in particular the set of escaping points in the boundary of $U$ has
harmonic measure zero. We also present some extensions of the results to the
case when $f$ has infinite degree on $U$, including classical Fatou example.
