The ergodic problem for some subelliptic operators with unbounded coefficients

We study existence and uniqueness of the invariant measure for stochastic processes with degenerate diffusion, whose infinitesimal gener- ators are linear subelliptic operators in the whole space R N with possibly unbounded coefficients. Such a measure together with a Liouville-type theorem will play a crucial role in two applications: the ergodic prob- lem studied through stationary problems with vanishing discount and the long time behavior of the solution to a parabolic Cauchy problem. In both cases, the constants will be characterized in terms of the invariant measure.