  This article rigorously analyzes the meeting time between pursuers and
evaders performing random walks on digraphs. There exist several bounds on the
expected meeting time between random walkers on graphs in the literature,
however, closed-form expressions are limited in scope. By utilizing the notion
that multiple random walks on a common graph can be understood as a single
random walk on the Kronecker product graph, we are able to provide the first
analytic expression for the meeting time in terms of the transition matrices of
the random walkers when modeled by either discrete-time Markov chains or
continuous-time Markov chains. We further extend the results to the case of
multiple pursuers and multiple evaders performing independent random walks. We
present various sufficient conditions for pairs (or tuples) of transition
matrices that satisfy certain conditions on the absorbing classes for which
finite meeting times are guaranteed to exist.
