  We consider Markov processes with generator of the form $\gamma
\mathcal{L}_{1} + \mathcal{L}_{0}$, in which $\mathcal{L}_{1}$ generates a
so-called dominant process that converges at large times towards a random point
in a fixed subset called the effective state space. Using the usual
characterization through martingales problems, we give general conditions under
which homogenization holds true: the original process converges, when $\gamma$
is large and for the Meyer-Zheng pseudo-path topology and for
finite-dimensional time marginals, towards an identified effective Markov
process on the effective space. Few simple model examples for diffusions are
studied.
