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.