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