  We study the Poisson equation Lu+f=0 in R^d, where L is the infinitesimal
generator of a diffusion process. In this paper, we allow the second-order part
of the generator L to be degenerate, provided a local condition of Doeblin type
is satisfied, so that, if we also assume a condition on the drift which implies
recurrence, the diffusion process is ergodic. The equation is understood in a
weak sense. Our results are then applied to diffusion approximation.
