Recurrence and Invariant Measures for Degenerate Diffusions

For a second-order (hypoelliptic) operator $\mathscr{A} = A_0 + \frac{1}{2} \sum^m_{i=1} A_i$ on a $d$-dimensional manifold $M^d$, let $x_t$ be the diffusion governed by $\mathscr{A}$ and $\varphi (t)$ its associated deterministic control system. We investigate the relations between transience, recurrence and (finite) invariant measures for $x_t$ using the control theoretic decomposition of $M^d$ with respect to $\varphi (t)$. On the invariant control sets for $\varphi (t)$ we obtain the same classification for $x_t$ as is well known for the nondegenerate case, while outside these sets the diffusion $x_t$ is transient.