Uniqueness in law for stochastic boundary value problems

We study existence and uniqueness of solutions for second order ordinary stochastic differential equations with Dirichlet boundary conditions on a given interval. In the first part of the paper we provide sufficient conditions to ensure pathwise uniqueness, extending some known results. In the second part we show sufficient conditions to have the weaker concept of uniqueness in law and provide a significant example. Such conditions involve a linearized equation and are of different type with respect to the ones which are usually imposed to study pathwise uniqueness. This seems to be the first paper which deals with uniqueness in law for (anticipating) stochastic boundary value problems. We mainly use functional analytic tools and some concepts of Malliavin Calculus.