Asymptotic stability and spiraling properties for solutions of stochastic equations

We consider a system of Itô equations in a domain in ${R^d}$. The boundary consists of points and closed surfaces. The coefficients are such that, starting for the exterior of the domain, the process stays in the exterior. We give sufficient conditions to ensure that the process converges to the boundary when $t \to \infty$. In the case of plane domains, we give conditions to ensure that the process ``spirals"; the angle obeys the strong law of large numbers.