Invariant measures associated to degenerate elliptic operators P. CannarsaGiuseppe Da PratoH. Frankowska 60H1047D0735K6537L40stochastic differential equationinvariancedegenerate elliptic operatorinvariant measure 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 $\mathbb{R}^n$. 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. Indiana University Mathematics Journal 2010 text pdf 10.1512/iumj.2010.59.3886 10.1512/iumj.2010.59.3886 en Indiana Univ. Math. J. 59 (2010) 53 - 78 state-of-the-art mathematics http://iumj.org/access/