Abstract: The boundary of the universal cover of a negatively curved manifold carries two natural measures: on the one hand, the Lebesgue (or Patterson-Sullivan) measure, on the other hand, hitting measures for random walks. We relate this problem to the dynamics of the geodesic flow on the manifold, in particular to the excursion into the cusp of random geodesics, and prove that such measures lie in different measure classes. Joint with Anja Randecker.
