Escaping points in the boundaries of Baker domains

We study dynamical behaviour of points in the boundaries of simply connected Baker domains U for meromorphic maps f, such that f has finite degree on U. We prove that if f on U has hyperbolic or simply parabolic type, then a typical point in the boundary of U (in the sense of harmonic measure) escapes to infinity under iteration. On the contrary, if f on U has doubly parabolic type, then a typical point in the boundary of U (in the sense of harmonic measure) is recurrent, in particular the set of escaping points in the boundary of U has harmonic measure zero. We also present some extensions to the case when f has infinite degree on U.