  We study a general molecular beam epitaxy (MBE) equation modeling the
epitaxial growth of thin films. We show that, in the deterministic case, the
associated Cauchy problem admits a unique smooth solution for all time, given
initial data in the space $X_0 = L^{2}(R^{d}) \cap \dot{W}^{1,4}(R^{d})$ with
$d = 1, 2$. This improves a recent result by Ag\'elas, who established global
existence in $H^{3}(R^{d})$. Moreover, we investigate the local existence and
uniqueness of solutions in the space $X_0$ for the stochastic MBE equation,
with an additive noise that is white in time and regular in the space variable.
