Ergodicity of regime-switching diffusions in Wasserstein distances

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 infinite countable state space are all studied. To deal with the infinite countable state space, we put forward a finite partition method. The boundedness for state-dependent regime-switching diffusions in an infinite countable state space is also studied.