  We study the large-time behavior of bounded from below solutions of parabolic
viscous Hamilton-Jacobi Equations in the whole space $\mathbb{R}^N$ in the case
of superquadratic Hamiltonians. Existence and uniqueness of such solutions are
shown in a very general framework, namely when the source term and the initial
data are only bounded from below with an arbitrary growth at infinity. Our main
result is that these solutions have an ergodic behavior when $t\to +\infty$,
i.e., they behave like $\lambda^*t + \phi(x)$ where $\lambda^*$ is the maximal
ergodic constant and $\phi$ is a solution of the associated ergodic problem.
The main originality of this result comes from the generality of the data: in
particular, the initial data may have a completely different growth at infinity
from those of the solution of the ergodic problem.
