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We address stability of a class of Markovian discrete-time stochastic hybridsystems. This class of systems is characterized by the state-space of thesystem being partitioned into a safe or target set and its exterior, and thedynamics of the system being different in each domain. We give conditions for$L_1$-boundedness of Lyapunov functions based on certain negative driftconditions outside the target set, together with some more minor assumptions.We then apply our results to a wide class of randomly switched systems (oriterated function systems), for which we give conditions for global asymptoticstability almost surely and in $L_1$. The systems need not be time-homogeneous,and our results apply to certain systems for which functional-analytic ormartingale-based estimates are difficult or impossible to get.

An Excursion-Theoretic Approach to Stability of Discrete-Time Stochastic Hybrid Systems