Invariance Principles

Abstract

In the previous two chapters, we have studied stability of switching diffusions and random switching ordinary differential equations. Continuing our effort, this chapter is concerned with invariance principles of switching diffusion processes. This chapter together with the previous two chapters delineates long-time behavior and gives a complete picture of the switchingdi ffusion processes under consideration.

The rest of the chapter is arranged as follows. Section 9.2 begins with the formulation of the problem. Section 9.3 is devoted to the invariance principles using sample paths and kernels of the associated Liapunov functions. Here, Liapunov function-type criteria are obtained first. Then linear (in x) systems are treated. Section 9.4 switches gears to examine the invariance using the associated measures. Finally, a few more remarks are made in Section 9.5 to conclude the chapter.