Stochastic Invariance and Degenerate Elliptic Operators

This paper is devoted to the invariance of a bounded open domain in Rn under a diffusion process with Lipschitz continuous data. We show that such a property holds under the same conditions that insure the invariance of a closed domain. This result is applied to get the well-posedness of a degenerate elliptic equation without imposing explicit boundary conditions, and to study the existence and uniqueness of the invariant measure for the associated transition semigroup. Key words: stochastic differential equation, invariance, degenerate elliptic operator, invariant measure. MSC Subject classifications: 60H10, 47D07, 35K65, 37L40.