  A stochastic timed automaton is a purely stochastic process defined on a
timed automaton, in which both delays and discrete choices are made randomly.
We study the almost-sure model-checking problem for this model, that is, given
a stochastic timed automaton A and a property $\Phi$, we want to decide whether
A satisfies $\Phi$ with probability 1. In this paper, we identify several
classes of automata and of properties for which this can be decided. The proof
relies on the construction of a finite abstraction, called the thick graph,
that we interpret as a finite Markov chain, and for which we can decide the
almost-sure model-checking problem. Correctness of the abstraction holds when
automata are almost-surely fair, which we show, is the case for two large
classes of systems, single- clock automata and so-called weak-reactive
automata. Techniques employed in this article gather tools from real-time
verification and probabilistic verification, as well as topological games
played on timed automata.
