  In this paper, we develop tools to establish almost sure stability of
stochastic switched systems whose switching signal is constrained by an
automaton. After having provided the necessary generalizations of existing
results in the setting of stochastic graphs, we provide a characterization of
almost sure stability in terms of multiple Lyapunov functions. We introduce the
concept of lifts, providing formal expansions of stochastic graphs, together
with the guarantee of conserving the underlying probability framework. We show
how these techniques, firstly introduced in the deterministic setting, provide
hierarchical methods in order to compute tight upper bounds for the almost sure
decay rate. The theoretical developments are finally illustrated via a
numerical example.
