Singular perturbations in ergodic control of diffusions

Ergodic control of a nondegenerate diffusion with two time-scales is studied in the limiting case as the time-scale separation increases to infinity. It is shown that the limit problem is another ergodic control problem for the slow time-scale component alone, with its dynamics averaged over the (controlled) invariant probability measures for the fast component. These measures in turn can be treated as the "effective control variable".