Invariant Measures for Hybrid Stochastic Systems
  In this paper, we seek to understand the behavior of dynamical systems that are perturbed by a parameter that changes discretely in time. If we impose certain conditions, we can study certain embedded systems within a hybrid system as time-homogeneous Markov processes. In particular, we prove the existence of invariant measures for each embedded system and relate the invariant measures for the various systems through the flow. We calculate these invariant measures explicitly in several illustrative examples. 
Xavier Garcia, Jennifer Kunze, Thomas Rudelius, Anthony Sanchez,   Sijing Shao, Emily Speranza, Chad Vidden
math.DS math.PR