Long time behavior of diffusions with Markov switching

Let Y be an Ornstein-Uhlenbeck diffusion governed by an ergodic finite state Markov process X: dYₜ=-λ(Xₜ)Yₜdt+σ(Xₜ)dBₜ, Y₀ given. Under ergodicity condition, we get quantitative estimates for the long time behavior of Y. We also establish a trichotomy for the tail of the stationary distribution of Y: it can be heavy (only some moments are finite), exponential-like (only some exponential moments are finite) or Gaussian-like (its Laplace transform is bounded below and above by Gaussian ones). The critical moments are characterized by the parameters of the model.