  We provide some criteria on the stability of regime-switching diffusion
processes. Both the state-independent and state-dependent regime-switching
diffusion processes with switching in a finite state space and an infinite
countable state space are studied in this work. We provide two methods to deal
with switching processes in an infinite countable state space. One is a finite
partition method based on the nonsingular M-matrix theory. Another is an
application of principal eigenvalue of a bilinear form. Our methods can deal
with both linear and nonlinear regime-switching diffusion processes. Moreover,
the method of principal eigenvalue is also used to study the recurrence of
regime-switching diffusion processes.
