  We derive the long time asymptotic of solutions to an evolutive
Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with
ergodic problems recently studied in \cite{bcr}. Our main assumption is an
appropriate degeneracy condition on the operator at the boundary. This
condition is related to the characteristic boundary points for linear operators
as well as to the irrelevant points for the generalized Dirichlet problem, and
implies in particular that no boundary datum has to be imposed. We prove that
there exists a constant $c$ such that the solutions of the evolutive problem
converge uniformly, in the reference frame moving with constant velocity $c$,
to a unique steady state solving a suitable ergodic problem.
