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 det,erniinistic subsystem. we study new intermediate situations. Tiye begin with the one-dimensional case that we essentially solve in full generality. We then consider the periodic case with constant coefiicients and show that ergodicity is equivalent to some stocliastic non resonance condition. Finall we show that the existence of one nondegenerate control is not sufficient for ergodicitj. in dimensions larger than or equal to 2

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