  Based on the theory of M-matrix and Perron-Frobenius theorem, we provide some
criteria to justify the convergence of the regime-switching diffusion processes
in Wasserstein distances. The cost function we used to define the Wasserstein
distance is not necessarily bounded. The continuous time Markov chains with
finite and countable state space are all studied. To deal with the countable
state space, we put forward a finite partition method. The boundedness for
state-dependent regime-switching diffusions in an infinite state space is also
studied.
