Abstract
This chapter provides characterisations of stochastic reachability thought of as a stochastic hybrid control problem. First, we define formally the stochastic reachability problem as a measure of those system trajectories that visit in finite or infinite time horizon a given target set. We prove that this concept is well defined. Then, we link stochastic reachability with other mathematical concepts, which come with richer theoretical and numerical characterisations. Such concepts are: hitting distributions, exit distributions, occupation measures, occupation time distribution, Choquet capacity and reduced function and balayage. These concepts belong to the Markov process equipment and each one is interesting by itself and has its own applications. These connections make stochastic reachability an extremely powerful and flexible tool that could have applications not only in verification problems, but also in problems like reliability, resilience, fault-tolerance, performability or other optimisation problems.
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