On unique ergodicity for degenerate diffusions

Abstract

We investigate the invariant probabilities of a possible degenerate diffusion process on a manifold. Using the support theorems of stroock, Varadhan and Kunita, the possible candidates for supports of invariant probabilities can be characterized as the invariant control sets of the corresponding control system. There remains the problem of how many invariant probabilities can coexist on one invariant the problem of how many invariant probabilities can coexist on one invariant control set C. Uniqueness on C is proved under the assumption that the Lie algebra generated by the drift and diffusion vector fields is full at one point in C. This generalizes the known results obtained by PDE methods. Several versions of the ergodic theorem are given.