  Regime switching processes have proved to be indispensable in the modeling of
various phenomena, allowing model parameters that traditionally were considered
to be constant to fluctuate in a Markovian manner in line with empirical
findings. We study diffusion processes of Ornstein-Uhlenbeck type where the
drift and diffusion coefficients $a$ and $b$ are functions of a Markov process
with a stationary distribution $\pi$ on a countable state space. Exact long
time behavior is determined for the three regimes corresponding to the expected
drift: $E_{\pi}a(\cdot)>0,=0,<0$, respectively. Alongside we provide exact time
limit results for integrals of form
$\int_{0}^{t}b^{2}(X_{s})e^{-2\int_{s}^{t}a(X_{r})dr}ds$ for the three
different regimes. Finally, we demonstrate natural applications of the findings
in terms of Cox-Ingersoll-Ross diffusion and deterministic SIS epidemic models
in Markovian environments. Exact long time behaviors are naturally expressed in
terms of solutions to the well-studied fixed-point equation in law
$X\stackrel{d}{=}AX+B$ with $X \indep (A,B)$.
