  This paper is devoted to the study of the large time behaviour of viscosity
solutions of parabolic equations with Neumann boundary conditions. This work is
the sequel of [13] in which a probabilistic method was developped to show that
the solution of a parabolic semilinear PDE behaves like a linear term $\lambda
T$ shifted with a function $v$, where $(v,\lambda)$ is the solution of the
ergodic PDE associated to the parabolic PDE. We adapt this method in finite
dimension by a penalization method in order to be able to apply an important
basic coupling estimate result and with the help of a regularization procedure
in order to avoid the lack of regularity of the coefficients in finite
dimension. The advantage of our method is that it gives an explicit rate of
convergence.
