  The purpose of the paper is to provide a general method for computing hitting
distributions of some regular subsets D for Ornstein-Uhlenbeck type operators
of the form 1/2\Delta + F\cdot\nabla, with F bounded and orthogonal to the
boundary of D. As an important application we obtain integral representations
of the Poisson kernel for a half-space and balls for hyperbolic Brownian motion
and for the classical Ornstein-Uhlenbeck process. The method developed in the
paper is based on stochastic calculus and on skew product representation of
multidimensional Brownian motion and yields more complete results as those
based on Feynmann-Kac technique.
