Singular Perturbations in Ergodic Control of Diffusions

Ergodic control of a non-degenerate diffusion with two timescales 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’.