  We introduce new sufficient conditions for verifying stability and recurrence
properties in singularly perturbed stochastic hybrid dynamical systems.
Specifically, we focus on hybrid systems with deterministic continuous-time
dynamics that exhibit multiple time scales and are modeled by constrained
differential inclusions, as well as discrete-time dynamics modeled by
constrained difference inclusions with random inputs. By assuming regularity
and causality of the dynamics and their solutions, respectively, we propose a
suitable class of composite nonsmooth Lagrange-Foster and Lyapunov-Foster
functions that can certify stability and recurrence using simpler functions
related to the slow and fast dynamics of the system. We establish the stability
properties with respect to compact sets, while the recurrence properties are
studied only for open sets.
