Feller continuity, recurrence, and stabilization of regime-switching diffusions

This work is based on our results in. Dealing with two-component Markov processes having continuous dynamics and discrete events, we develop asymptotic properties of regime-switching diffusions. First, strong Feller property is established. Next, classifying the underlying processes as recurrence (positive recurrence or null recurrence) or transience, sufficient conditions for such processes are obtained. In addition to providing general criteria for recurrent and transient switching diffusions, for processes that are linearizable with respect to the continuous component, we establish easily verifiable conditions. Furthermore, we provide conditions for controlled switching diffusions in which the feedback controls ensure weak stability (or ergodicity). Finally, we give a simple example to demonstrate the results.