  This article surveys recent advancements of strategy designs for persistent
robotic surveillance tasks with the focus on stochastic approaches. The problem
describes how mobile robots stochastically patrol a graph in an efficient way
where the efficiency is defined with respect to relevant underlying performance
metrics. We first start by reviewing the basics of Markov chains, which is the
primary motion model for stochastic robotic surveillance. Then two main
criteria regarding the speed and unpredictability of surveillance strategies
are discussed. The central objects that appear throughout the treatment is the
hitting times of Markov chains, their distributions and expectations. We
formulate various optimization problems based on the concerned metrics in
different scenarios and establish their respective properties.
