  We study the initial-boundary value problem for the Fokker-Planck equation in
an interval with absorbing boundary conditions. We develop a theory of
well-posedness of classical solutions for the problem. We also prove that the
resulting solutions decay exponentially for long times. To prove these results
we obtain several crucial estimates, which include hypoellipticity away from
the singular set for the Fokker-Planck equation with absorbing boundary
conditions, as well as the Holder continuity of the solutions up to the
singular set.
