On ergodic stochastic control

We study here ergodic optimal stochastic control problems. After recalling some "classical" cases where the control system is known to be ergodic like the uniformly nondegenerate case or when there is an exactly controllable deterministic subsystem, we study new intermediate situations. We begin with the one-dimensional case that we essentially solve in full generality. We then consider the periodic case with constant coefficients and show that ergodicity is equivalent to some "stochastic non resonance condition". Finally, we show that the existence of one nondegenerate control is not sufficient for ergodicity in dimensions larger than or equal to 2.