Let Y be a Ornstein–Uhlenbeck diffusion governed by a
stationary and ergodic process X : dYt = a(Xt)Ytdt + σ(Xt)dWt,Y0 = y0.
We establish that under the condition α = Eµ(a(X0)) < 0 with μ the stationary distribution of
the regime process X, the diffusion
Y is ergodic.
We also consider conditions for the
existence of moments for the
invariant law of Y when X is a Markov jump process
having a finite number of states.
Using results on random difference equations
on one hand and the fact that conditionally to
X, Y is Gaussian on the other hand,
we give such a condition for the existence of
the moment of order s ≥ 0. Actually we recover in this case
a result that
Basak et al. [J. Math. Anal. Appl. 202 (1996) 604–622]
have established using the theory of stochastic control of
linear systems.