Cusp excursion in hyperbolic manifolds and singularity of harmonic measure

We generalize the notion of cusp excursion of geodesic rays by introducing for any k ≥ 1 the kth excursion in the cusps of a hyperbolic N manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk. On the other hand, for k = N − 1, the kth excursion is superlinear for geodesics that are generic with respect to the Lebesgue measure. We use this to show that the hitting measure and the Lebesgue measure on the boundary of hyperbolic space ℍN for any N ≥ 2 are mutually singular.