Invariant measures associated to degenerate elliptic operators

This paper is devoted to the study of the existence and uniqueness of the invariant measure associated to the transition semigroup of a diffusion process in a bounded open subset of Rn. For this purpose, we investigate first the invariance of a bounded open domain with piecewise smooth boundary showing that such a property holds true under the same conditions that insure the invariance of the closure of the domain. A uniqueness result for the invariant measure is obtained in the class of all probability measures that are absolutely continuous with respect to Lebesgue’s measure. A sufficient condition for the existence of such a measure is also provided.