  This paper analyzes the meeting time between a pair of pursuer and evader
performing random walks on digraphs. The existing bounds on the meeting time
usually work only for certain classes of walks and cannot be used to formulate
optimization problems and design robotic strategies. First, by analyzing
multiple random walks on a common graph as a single random walk on the
Kronecker product graph, we provide the first closed-form expression for the
expected meeting time in terms of the transition matrices of the moving agents.
This novel expression leads to necessary and sufficient conditions for the
meeting time to be finite and to insightful graph-theoretic interpretations.
Second, based on the closed-form expression, we setup and study the
minimization problem for the expected capture time for a pursuer/evader pair.
We report theoretical and numerical results on basic case studies to show the
effectiveness of the design.
