  We study the ergodic behaviour of a discrete-time process $X$ which is a
Markov chain in a stationary random environment. The laws of $X_t$ are shown to
converge to a limiting law in (weighted) total variation distance as
$t\to\infty$. Convergence speed is estimated and an ergodic theorem is
established for functionals of $X$.
  Our hypotheses on $X$ combine the standard "small set" and "drift" conditions
for geometrically ergodic Markov chains with conditions on the growth rate of a
certain "maximal process" of the random environment. We are able to cover a
wide range of models that have heretofore been untractable. In particular, our
results are pertinent to difference equations modulated by a stationary
Gaussian process. Such equations arise in applications, for example, in
discretized stochastic volatility models of mathematical finance.

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