We employ set oriented methods in combination with graph partitioning algorithms to identify key dynamical regions in the Sun-Jupiter-particle three-body system. Transport rates from a region near the 32 Hilda resonance into the realm of orbits crossing Mars' orbit are computed. In contrast to common numerical approaches, our technique does not depend on single long term simulations of the underlying model. Thus, our statistical results are particularly reliable since they are not affected by a dynamical behavior which is almost nonergodic (i.e., dominated by strongly almost invariant sets).