  For a wide class of continuous-time Markov processes, including all
irreducible hypoelliptic diffusions evolving on an open, connected subset of
$\RL^d$, the following are shown to be equivalent: (i) The process satisfies (a
slightly weaker version of) the classical Donsker-Varadhan conditions; (ii) The
transition semigroup of the process can be approximated by a finite-state
hidden Markov model, in a strong sense in terms of an associated operator norm;
(iii) The resolvent kernel of the process is `$v$-separable', that is, it can
be approximated arbitrarily well in operator norm by finite-rank kernels. Under
any (hence all) of the above conditions, the Markov process is shown to have a
purely discrete spectrum on a naturally associated weighted $L_\infty$ space.
