Escaping points in the boundaries of Baker domains

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 ofU,with respect to harmonicmeasure, escapes to infinity under iteration of f. 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 the classical Fatou example.