Positive Recurrence: Weakly Connected Ergodic Classes

Abstract

To study the positive recurrence and ergodicity, one of the conditions used in Chapters 3 and 4 is that the states of the switching process belong to only one ergodic class. In this chapter, we further our study by treating a more general class of problems. We consider the case that the states of the discrete event process belong to several “ergodic” classes that are weakly connected. This notion is made more precise in what follows. A key idea is the use of two-time-scale formulation; see [176, 177] and many references therein.

The rest of the chapter is arranged as follows. Section 10.2 begins with the formulation. Section 10.3 focuses on hybrid diffusions whose discrete component lives in weakly connected “ergodic” (irreducible) classes. Finally, the chapter is concluded with additional remarks in Section 10.4.