In this paper we study asymmetric proximity measures on directed graphs, which quantify the relationships between two nodes or two groups of nodes. The measures are useful in several graph mining tasks, including clustering, link prediction and connection subgraph discovery. Our proximity measure is based on the concept of escape probability. This way, we strive to summarize the multiple facets of nodes-proximity, while avoiding some of the pitfalls to which alternative proximity measures are susceptible. A unique feature of the measures is accounting for the underlying directional information. We put a special emphasis on computational efficiency, and develop fast solutions that are applicable in several settings. Our experimental study shows the usefulness of our proposed direction-aware proximity method for several applications, and that our algorithms achieve a significant speedup (up to 50,000x) over straightforward implementations.