A general martingale approach to large noise homogenization

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.